fornj 
nal 


THE  LIBRARY 

OF 

THE  UNIVERSITY 
OF  CALIFORNIA 

LOS  ANGELES 


GIFT  OF 

•John  S.Prpll 


PI.YYVJTT 


INDUSTRIAL  DRAWING 


COMPRISING 


THE    DESCRIPTION  AND  USES 


DRAWING    INSTRUMENTS, 

THE  CONSTRUCTION  OF  PLANE  FIGURES, 
TINTING, 

THE  PROJECTIONS  AND    SECTIONS   OF  GEOMETRICAL    SOLIDS,   SHADOWS, 

SHADING,    ISOMETRICAL   DRAWING,   OBLIQUE  PROJECTION, 

PERSPECTIVE,     ARCHITECTURAL    ELEMENTS, 

MECHANICAL  AND  TOPOGRAPHICAL  DRAWING. 

FOR  THE  USE  OP  HIGH  SCHOOLS,  ACADEMIES,  AND  SCIENTIFIC  SCHOOLS, 
BY  D.  H.  MAHAN,  LL.D., 

LATE  PROFESSOR   OF  CIVIL  ENGINEERING,  ETC.,   IN   THE  UNITED  STATES 
MILITABY   ACADEMY. 

REVISED    AND     ENLARGED 

BY 

DWINEL    F.    THOMPSON,    B.S., 

PBOFESSOB   OF  DESCRIPTIVE  GEOMETRY,    STEREOTOMY,    AND    DRAWING    IN    THE 
RENSSELAEB   POLYTECHNIC  INSTITUTE,    TBOY,    N.   Y. 


NEW  YORK: 
JOHN  WILEY  &  SONS, 

i  5  ASTOP   PLAGE. 
1888. 

JOHN  S.  PRELL 

Goil  &  Mechanical  Engineer. 

*.U\T   FU  A.M.,  I  .SCO,  <J.\U 


COPTBIQHT    B» 

JOHN    WILEX. 
1&J7. 


Engineering 
Library 


M 


PREFACE. 


THIS  revised  edition  has  been  enlarged  by  the  addition  of 
the  chapters  on  Tinting,  Shadows,  Shading,  Isometrical  Draw- 
ing, Oblique  Projection,  and  Perspective.  The  chapters  on 
Drawing  Instruments  and  their  Uses  have  been  rewritten  and 
much  new  matter  added ;  while  some  changes  and  additions 
have  been  made  in  the  chapters  on  Projections  and  Topo- 
graphy. 

It  is  hoped  that  by  these  changes  the  book  will  prove  more 
useful  in  the  class-room,  or  as  a  guide  for  self -instruction. 

The  cuts  of  Drawing  Instruments  used  in  this  work, 
marked  K.  &  E.  in  the  table  of  contents,  have  been  kindly 
furnished  by  Keuffel  &  Esser,  No.  Ill  Fulton  St.,  New  York. 

D.  F.  T. 


733392 


CONTENTS. 


CHAPTER  I. 

DRA"WTNQ  INSTRUMENTS  AND  MATERIALS. 

PAGE 

Compasses  (K.  &  E.} 1 

Hair-spring  dividers  (K.  &  E.) 2 

Bow  compasses  (K.  &  E.) 3 

Proportional  compasses  (K.  &  E.) 3 

Beam  compasses  (K.  &  E.) 4 

Drawing  pen  (K.  &  E.) 5 

Drawing1  board  (K.  &  E.) 5 

Tsquare(jST.  &  E.) 6 

Triangles  (JC  &  E.) 7 

Irregular  curves 7 

India  ink 8 

Colors 8 

Ink  saucers 8 

Pencils 8 

Brushes 8 

India  rubber 8 

Thumb-tacks 8 

Horn  centre 8 

Drawing  paper. 9 

Tracing  paper  and  cloth 9 

Scales 9 

A  scale  of  equal  parts 10 

Manner  of  using  the  scale 11 

Diagonal  scale  of  equal  parts 13 

Duodecimally  divided  scales  (K.  &  E.) 17 

Chain  scales 17 

Protractor  (K.  <t  E.) 18 

CHAPTER  II. 

USE  AND  CAKE  OP  INSTRUMENTS. 

Compasses 19 

Dividers 19 

Drawing  pen 20 

Sharpening  pen. 20 


Vi  CONTENTS. 

FAGI 

Drawing  board < 21 

T  square 22 

Triangles 23 

Manner  of  using  triangles,  etc. 23 

Irregular  curves 25 

Preparation  of  ink 25 

Pencils 26 

Stretching  paper 26 

Pencilling  a  drawing 27 

Juicing  a  drawing • 27 

Accuracy. 28 

Cleaning  a  drawing 28 

Cuttingoff  adrawing 29 

Tracing. 29 

Accuracy  of  scales 29 


CHAPTER  III. 

CONSTRUCTION  OP  PROBLEMS  OP  POINTS  AND  STRAIGHT  LINES. 

Prob.    1. — To  draw  a  straight  line  through  two  given  points 31 

44      2. — To  set  off  a  given  distance  along  a  straight  line  from  a  point 

on  it 33 

44      3. — To  set  off  along  a  straight  line  any  number  of  equal  dis- 


4.  — From  a  given  point  on  a  right  line  to  set  off  any  number  of 

unequal  distances 34 

5. — To  divide  a  given  line  into  equal  parts 35 

6. — To  construct  a  perpendicular  to  a  line  from  a  point  on  it. . .  36 

7. — To  construct  a  perpendicular  to  a  line  at  its  extremity 38 

8. — To  construct  a  perpendicular  to  a  line  from  a  point  not  on  it.  39 
9. — To  construct  a  perpendicular  to  a  line  from  a  point  without 

near  its  extremity 39 

"    10.— To  set  off  a  given  distance  from  a  line 40 

44    11.— To  draw  a  parallel  to  a  line 40 

41     12. — To  draw  a  parallel  to  a  line  at  a  given  distance  from  it 41 

"     13. — To  transfer  an  angle 41 

44    14.— To  construct  an  angle  of  60° 41 

"    15. — To  construct  an  angle  of  45° 42 

44     16.— To  bisect  an  angle 42 

44    17. — To  bisect  the  angle  between  two  lines  that  do  not  meet. ...  42 

Construction  of  Ares  of  Circles,  Straight  Lines  and  Points. 

Prob.  18.  — To  describe  an  arc  with  a  given  radius  through  two  points . .  43 

44    19.  —To  find  the  centre  of  a  circle  described  through  three  points.  43 

44    20. — To  construct  a  tangent  to  an  arc 44 

"    21. — To  construct  an  arc  of  a  given  radius  tangent  to  a  given  arc.  45 


CONTENfS.  Vil 

PAGE 

Prob.  22. — To  construct  a  tangent  to  an  arc  from  a  point  without 45 

"  23.— To  draw  a  tangent  to  two  circles 46 

"  24. — To  construct  a  circle  with  a  given  radius  tangent  to  two  lines.  46 
"  25.— To  construct  any  number  of  circles  tangent  to  each  other 

and  to  two  lines 47 

"  26. — To  construct  a  given  circle  tangent  to  another  and  to^a  right 

line .' 47 

44  27. — To  construct  a  circle  tangent  to  another  at  a  given  point  and 

to  a  right  line 48 

44  28. — To  construct  a  circle  tangent  to  another  and  to  a  right  line 

at  a  given  point. 48 

"  29.— To  construct  a  circle  tangent  to  two  others 49 

"  30.— To  construct  a  half  oval  of  three  centres 49 

*'  81. — To  construct  a  four  centre  curve 50 

"  32.— To  construct  a  half  oval  of  five  centres. 51 

"  33.— To  construct  a  quarter  of  a  three  centre  oval  tangent  to  two 

lines 53 

44  34. — To  constrict  a  quarter  five  centre  oval  tangent  to  two  lines.  53 

"  35.— To  construct  an  S  curve  tangent  to  two  parallel  lines 53 

44  36. — To  construct  an  S  curve  having  its  centres  on  two  parallel 

lines. 53 

Construction  of  Problems  of  Circles  and  Rectilineal  Figures. 

Prob.  37. — To  construct  a  triangle 54 

44    38. — To  construct  a  square 54 

44    39. — To  construct  a  parallelogram 54 

44    40. — To  circumscribe  a  triangle  by  a  circle 55 

41. — To  inscribe  a  circle  in  a  triangle     55 

42. — To  inscribe  a  square  in  a  circle 55 

43. — To  inscribe  an  octagon  in  a  circle 55 

44, — To  inscribe  a  hexagon  in  a  circle 56 

4    45. — To  inscribe  an  equilateral  triangle  in  a  circle 56 

"    46,— To  inscribe  a  pentagon  in  a  circle 56 

44    47. — To  construct  a  regular  figure  of  a  given  side  (two  methods) . .  57 

44    48. — To  circumscribe  a  circle  by  a  regular  figure 58 

44    49. — To  inscribe  a  circle  in  a  regular  figure 58 

44    50. — To  inscribe  equal  circles  in  a  given  circle 58 

44    51. — To  circumscribe  a  circle  by  equal  circles 69 

Construction  of  Proportional  Lines  and  Figures. 

Prob.  52.— To  divide  a  line  into  two  proportional  parts 60 

44     53. — To  divide  a  line  into  any  number  of  parts  proportional  to  each 

other 60 

44    54. — To  construct  a  fourth  proportional 60 

i4    55. — To  construct  a  mean  proportional 61 

44    56. — To  divide  a  line  into  mean  and  extreme  ratio 61 

44    57. — To  construct  a  figure  proportional  to  another 62 


Viii  CONTENTS. 

Construction  of  Equivalent  Figures. 

PAOB 

Prob.  58  — To  construct  a  triangle  equivalent  to  a  parallelogram 62 

"     59. — To  construct  a  triangle  equivalent  to  a  quadrilateral 62 

"    60. — To  construct  a  triangle  equivalent  to  a  polygon 63 

"    61. — To  construct  a  triangle  equivalent  to  a  regular  polygon 63 

'.          Construction  of  Curved  Lines  by  Points. 

Prob.  62. — To  construct  an  ellipse  (two  methods) 63 

'    63. — To  construct  an  ellipse  with  three  points  given 65 

'    64. — To  construct  a  tangent  to  an  ellipse  at  a  given  point 66 

'    65. — To  construct  a  tangent  to  an  ellipse  from  a  point  without. . .  66 

'    66. — To  copy  a  curve  by  points 67 

*    67. — To  make  a  copy  greater  or  smaller  than  a  given  figure 67 

'    68. — To  describe  an  arc  of  a  circle  by  points 68 

'    69. — To  construct  a  parabola  by  points 68 

'    70. — To  construct  the  circumference  of  a  circle  of  given  diameter.  69 

CHAPTER  IV. 

TINTING  AND  SHADING. 

Flat  tints,  line  shading 70 

Graduated  tints,  line  shading 71 

Flat  tints,  India  ink. 71 

Graduated  tints,  India  ink 72 

Dry  shading. 73 

Colors 74 

CHAPTER  V. 

CONVENTIONAL  MODES  OP  REPRESENTING  DIFFERENT  MATERIALS. 

Conventional  tints 75 

Wood , 75 

Masonry 77 

Metals 78 

Earth, 78 

Water 78 

CHAPTER  VI. 

CONSTRUCTION   OF  REGULAR  FIGURES. 

To  represent  a  pavement  made  up  of  squares , 79 

To  represent  a  pavement  of  equilateral  triangles 79 

To  represent  a  pavement  of  hexagons 79 

To  represent  a  pavement  made  up  of  octagons  and  squares 79 

To  represent  a  pavement  made  up  of  isosceles  triangles 80 

Examples  of  architectural  ornament .80 


CONTENTS.  U 

CHAPTER  VII. 
PBOJECTIOKa 

PAGE 

Principles. 82 

Notation. 86 

Shade  lines 86 

Profiles  and  sections 87 

Projections  of  points  and  right  lines 89 

Prob.   71. — To  construct  the  projections  of  a  right  pyramid 92 

' '      72.  — To  construct  the  parts  of  a  right  pyramid  from  its  projections    93 

"      73. — To  construct  the  projections  of  an  oblique  pyramid 93 

"      74. — To  construct  the  projections  of  a  right  prism 94 

Traces  of  planes  on  the  planes  of  projection 95 

Prob.  75.  — To  construct  the  projections  and  sections  of  a  hollow  cube.  96 
"  76. — To  construct  the  projections  and  sections  of  a  hollow  pyramid  98 

"      77. — To  construct  the  plans,  elevations,  &c.,  of  a  house 102 

Cylinder 106 

Prob.   85. — Projections  of  the  cylinder 107 

"      86.— The  cone  and  its  projections 108 

"      87.— The  sphere  and  its  projections 109 

"      88.— Oblique  section  of  the  cylinder  (two  methods) 109 

"      89. — Oblique  sections  of  the  cone  (three  cases) Ill 

"      90. — Oblique  sections  of  the  cone  (two  cases) 112 

"      91.— Section  of  the  sphere 112 

"      92. — To  construct  the  curves  of  section  of  the  cylinder,  cone  and 

sphere  (1st  method) 113 

"      93. — Second  method  of  preceding  problem 114 

Projections  of  cylinder  and  cones  with  axes  oblique  to  the  planes  of 

projection 115 

Preliminary  problems  94  and  95 115-117 

Prob.  96.— Projections  of  cylinder  with  axis  oblique  to  vertical  plane.  118 
"  97. — Projections  of  cylinder  with  axis  oblique  to  horizontal  plane.  119 
"  98. — Projections  of  cylinder  with  axis  oblique  to  both  planes  of 

projection. 119 

"      99. — Projections  of  cone  with  axis  parallel  to  vertical  plane 121 

"     100. — Projections  of  cone  with  axis  oblique  to  vertical  plane 122 

"    101. — Projections  of  cone  with  axis  oblique  to  both  planes  of  pro- 
jection  122 

"    102.— Projections  of  hollow  cylinder  with  axis  parallel  to  vertical 

plane 122 

"    103. — Projections  of  hollow  cylinder  with  axis  oblique  to  both 

planes 123 

"    104. — Projections  of  hollow  hemisphere  (1st  case) 123 

"    105. — Projections  of  hollow  hemisphere  (2d  case) 124 

Intersections  of  Cylinders,  Cones,  and  Spheres. 

Prob.  106..— Intersection  of  two  right  cylinders  with  axes  parallel  to 

vertical  plane 125 


CONTENTS. 


Prob.  107.— Intersection  of  two  right  cylinders,  the  axis  of  one  oblique 

to  horizontal  plane 127 

"    108. — Intersection  of  two  right  cylinders,  the  axis  of  one  oblique 

to  both  planes. 128 

"    109.— Intersection  of  cone  and  cylinder 128 

"    110. — Intersection  of  cylinder  and  hemisphere 130 

"    111.— Intersection  of  cone  and  sphere 131 

Development  of  Cylindrical  and  Conical  Surfaces. 

Prob.  112.— Development  of  cylinder 133 

"    113.— Development  of  cone 134 

CHAPTER  VIII. 

SHADOWS. 

Principles 1 36 

Prob.  .1.— To  find  the  shadow  of  a  cube , 137 

"      2. — To  find  the  shadow  of  the  frustum  of  a  square  pyramid. . . .  137 

"      3. — To  find  the  shadow  of  a  short  hexagonal  prism 137 

"      4.— To  find  the  shadow  of  a  vertical  cylinder 137 

"      5.  — Shadow  of  a  cylindrical  abacus  upon  an  octagonal  prism ....  138 

"      6. — Shadow  on  interior  of  hollow  semi-cylinder. 188 

"      7.— Shadow  on  steps 139 

"      8.— Shadow  of  framing 139 

"      9. — Shadow  of  timber  resting  upon  top  of  wall 140 

"     10. — Shadow  of  inclined  timber  upon  triangular  prism 140 

"     11. — To  find  the  angle  which  a  ray  of  light  makes  with    .  Jher 

plane  of  projection 141 

"     12. — Shadow  on  interior  of  hollow  hemisphere 141 

Line  of  shade 141 

Prob.  13.— To  find  the  line  of  shade  upon  a  cone 142 

"    14.— To  find  the  line  of  shade  upon  a  sphere 142 

CHAPTEE  IX. 

SHADING. 

Rules. 144 

Application  of  rules  to  the  shading  of  an  hexagonal  prism 144 

«•  "          "  "  acylinder 145 

"          "  "  a  cone  and  sphere 145 

CHAPTER  X. 

ISOMETRICAL  DRAWING, 

Principles  of  isometrical  projection 146-148 

Prob.    1. — To  construct  the  isometrical  drawing  of  a  cube,  with  a  block 

upon  one  face  and  a  recess  in  another 148 


CONTENTS.  XI 

PAOB 
Prob.  2.— To  construct  the  isometrical  drawing  of  three  pieces  of 

timber  bolted  together 149 

*  •  3. — To  construct  the  isometrical  drawing  of  a  portion  of  framing.  149 

*'  4. — To  make  the  isometrical  drawing  of  a  circle 149 

"  5. — To  make  an  approximate  construction  of  the  isometrical 

drawing  of  a  circle 150 

"  6. — To  divide  the  isometrical  drawing  of  a  circle  into  equal  parts.  150 
**  7. — To  make  the  isometrical  drawing  of  a  cube,  cylinder,  and 

sphere 150 

•'  8. — Isometrical  drawing  of  brackets  supporting  a  shelf 151 

"  9. — The  isometrical  drawing  of  a  nut  and  washer. 151 

"  10.— The  isometrical  drawing  of  letters. 153 

Shadows 152 

Prob.  11. — The  isometrical  drawing  of  a  cube  with  its  shadow  upon  the 

horizontal  plane 153 

"  12. — The  isometrical  drawing  of  an  hexagonal  prism  with  the 

shadow 153 

"  13.— The  isometrical  drawing  of  a  beam,  projecting  from  a  verti- 
cal wall,  with  the  shadow 153 

"  14. — The  isometrical  drawing  of  a  four-armed  cross,  with  the 

shadow 153 

"  15. — The  isomefcrical  drawing  of  a  vertical  cylinder,  passing 

through  an  hexagonal  block 154 


CHAPTER  XI. 

OBLIQUE  PROJECTION. 

Principles  of  oblique  projection 155 

Prob.  1. — To  make  the  oblique  projection  of  a  circle i 156 

CHAPTER  XII. 

LINEAR   PERSPECTIVE. 

Principles  of  perspective 158 

Prob.  1. — To  find  the  perspective  of  right  lines  in  different  positions  by 

means  of  visual  rays 159 

"  2. — To  find  the  perspective  of  a  cube  when  placed  with  a  face 

paralleltoV 159 

"  3. — To  find  the  perspective  of  a  cube  when  placed  with  its  faces 

oblique  to  V  160 

"  4. — To  construct  the  perspective  of  a  regular  hexagon  by  means 

of  diagonals  and  perpendiculars 162 

"  5. — To  construct  the  perspective  of  a  pavement  made  up  of 


6. — To  find  the  perspective  of  a  cube 

7. — To  find  the  perspective  of  a  vertical  hexagonal  prism 


xii  CONTENTS. 

MM 

Prob.    8.— To  find  the  perspective  of  a  square  pillar  resting  upon  a 

pedestal 163 

«      9.— To  find  the  perspective  of  a  square  pyramid 164 

«««    10.— To  find  the  perspective  of  a  square  pyramid  resting  upon  a 

pedestal 164 

•*    11. — To  find  the  perspective  of  an  hexagonal  prism,  whose  axis  is 

parallel  to  H  and  inclined  to  V 164 

"    12.— To  find  the  perspective  of  a  circle 165 

•*    13. — To  find  the  perspective  of  a  circle  when  it  is  perpendicular 

to  both  H  and  V 1 65 

««    14 — To  find  the  perspective  of  a  vertical  cylinder 165 

"    15.— To  find  the  perspective  of  a  cylinder  whose  axis  is  perpendi- 
cular to  V 1!vS 

"    16.— To  find  the  perspective  of  a  rectangular  block  with  a  semi- 
circular top 166 


CHAPTER  Xin. 

ARCHITECTURAL  ELEMENTS. 

Roof  truss 168 

Columns  and  entablatures 169 

Arches... 171 


CHAPTER  XIV. 

MECHANISM. 

Prob.  114. — Projections  of  cylindrical  spur  wheel 175 

"  115. — Projections  of  cylindrical  spur  wheel  with  axis  oblique  to 

vertical  plane 177 

"  116.— Projections  of  mitre  wheel. 177 

"  117.— Projections  of  mitre  wheel  with  axis  oblique  to  vertical  plane.  180 

"  118.— Projections  of  a  helix  on  a  cylinder 180 

"  119.— Projections  of  screw  with  square  fillet. 182 

"  120. — Projections  and  sections  of  the  mechanism  of  a  part  of  a 

steam  engine 183 

"  121.— Sketches  and  finished  drawings  of  the  crab  engine. 184 

CHAPTER  XV. 

TOPOGRAPHICAL  DRAWING. 

Preliminary  remarks  on  the  manner  of  representing  irregular  surfaces.  187 

Plane  of  comparison 191 

References 192 

Projections  of  horizontal  curves 192 

Pi-ob.  123.— To  construct  the  horizontal  curves  from  a  survey 194 


CONTENTS.  X1U 

Conventional  Methods  of  representing  the  Natural  and  Artificial  Fea- 
tures of  a  Locality. 

PAGE 

Slopes  of  ground. 197 

Scale  of  spaces 200 

Surfaces  of  water 201 

Shores. 201 

Meadows,  marshy  grounds,  trees,  rivulets,  rocks,  artificial  objects 202 

Practical  methods. 203 

Colored  topography 203 

Scales  of  distances 206 

Table  of  scales ...  207 

Copying  maps 208 


CHAPTER  1. 

DBA  WING   INSTRUMENTS   AND   MATERIALS. 

1  Good  instruments  are  necessary  for  good  work,  and  the 
beginner  who  desires  to  excel  should  procure  the  best.  These 
can  be  purchased  by  the  case,  or  by  the  piece.  The  latter 
being  the  best  way,  as  there  are  sometimes  pieces  in  the  case 
which  are  seldom,  if  ever,  used.  After  selecting  the  pieces  a 
case  can  be  made  for  them,  or,  if  one  does  not  wish  to  incur 
that  expense,  roll  them  up  in  a  piece  of  chamois  skin,  arrang- 
ing so  that  they  do  not  touch  each  other.  By  folding  over 
one  edge  of  the  skin,  and  stitching  together,  little  pockets  may 
be  formed  for  each  piece. 

The  following  instruments  will  be  necessary :  a  compass 
with  pen,  pencil  and  needle  points,  and  lengthening  bar  ;  hair- 
spring dividers ;  bow  compass  with  pen  and  pencil  points ;  and 
one  or  two  drawing  pens.  These  with  a  drawing  board,  T 
square,  triangles,  etc.,  will  make  a  sufficient  outfit. 

2.  Compasses.    These  instruments  are  so  well  known  as 
hardly  to  require  description  here.     They  consist  of  two  legs, 
which  are  connected  by  a  joint  or  hinge.     The  extremities  of 
the  legs  are  finished  with  fine  steel  points,  the  upper  portions 
are  of  brass,  German  silver,  or  some  other  metal  that  does  not 
readily  rust.     The  joint  is  generally  made  partly  of  steel,  as 
the  wear  will  be  more  equal.     The  joint  should  be  accurate 
and  firm,  but  admit  of  easy  play  without  any  sudden  jerks  in 
opening  and  closing  the  legs.     If  by  use  the  joint  becomes 
loose,  it  can  be  tightened  with  the  small  screw-driver  which 
accompanies  the  instruments.     The  points  should  come  accu- 
rately together  when  the  legs  are  closed. 

3.  Compass  with  movable  parts.    Fig.  1.  represents  a  good 
form  of  this  instrument,  with  its  furniture,  the  pen,  pencil, 


2 


INDUSTRIAL  DRAWING. 


and  needle  points,  and  lengthening  bar.  This  form  of  needle 
point  is  much  the  best  ;  the  shoulder  of  the  needle  should  be 
large  enough  to  prevent  making  a  hole  in  the  paper  when 
used.  This  style  of  pencil  point  is  also  very  convenient,  being 
made  for  the  use  of  leads.  The  lengthening  bar  is  used  in 
describing  larger  circles  than  the  compasses  would  otherwise 
admit  of.  With  some  of  these  instruments  there  is  a  dotting 
pen,  but  this  is  not  recommended  as  it  cannot  be  depended 
upon.  In  selecting,  see  that  the  parts  all  fit  well  together. 


Fio.  1 


FIG.  a 


4.  Hair-spring  dividers.  These  are  so  constructed,  that 
by  means  of  a  spring  and  screw,  the  distance  between  the 
points  may  be  changed  without  any  motion  at  the  joint.  By 
turning  the  screw,  Fig.  2,  the  point  may  be  moved  a  very  fine 
distance,  thus  making  it  very  convenient  for  taking  measure- 


DRAWING  INSTRUMENTS   AND   MATERIALS.  3 

ments  from  scales,  or  dividing  distances  into  equal  j>arts. 
The  spring  should  be  pretty  stiff. 

5.  Bow  compasses.  These  are  for  drawing  small  circles 
both  in  pencil  and  ink.  There  are  some  forms  made,  where 
the  pen  and  pencil  points  can  be  exchanged  as  in  the  larger 


compasses.  It  is  more  convenient,  however,  to  have  ,«> 
separate  instruments,  one  for  the  pen,  and  the  other  foi  che 
pencil,  as  shown  in  Fig.  3.  See  that  the  springs  are  stroig. 

There  are  other  forms  of  compasses,  which  are  convenient 
at  times,  but  not  necessary  to  buy  at  first.  Among  thei,e  are 
the  proportional  and  beam  compasses. 

6.  Proportional  compasses.  These  are  useful  for  en"  urging 
or  reducing  drawings.  The  simplest  form,  shown  ifc  Fig.  4, 
is  called  halves  and  wholes ;  where  the  shorter  legs  .*re  half 
the  length  of  the  others,  so  that  the  distance  between  the 
points  of  the  shorter  legs  will  be  one  half  of  that  bet  tveen  the 
points  at  the  other  end. 

Fig.  5  shows  another  form  of  these  compasses,  &  tmilar  in 
principle  to  the  last,  but  constructed  so  that  the  position  of 
the  joint  can  be  changed,  thus  giving  different  p.  ©portions 
between  the  extremities  at  either  end.  To  adjust  the  instru- 
ment, close  it,  and  after  unscrewing  the  nut,  move  the  slide 


INDUSTRIAL   DRAWING. 


along  until  the  mark  across  it  coincides  with  the  required 
number,  then  clamp  the  nut. 

Four  scales  are  sometimes  engraved  upon  these  compasses, 
called  lines,  circles,  planes,  and 
solids.  The  last  two  are  omitted 
upon  many  of  the  instruments,  as 
they  are  of  little  use. 

If  the  mark  is  brought  to  £  in 
the  scale  of  lines,  the  distance  ab 
will  be  one-third  of  cd.  If  the 
mark  is  brought  to  8  in  the  scale  of 
circles,  ab  will  be  the  chord  of  the 
eighth  part  of  the  circumference 
of  a  circle  whose  radius  is  cd. 
Place  the  mark  at  4  in  the  scale  of 
planes,  and  ab  will  be  the  side  of 
a  square,  or  radius  of  a  oircle 
whose  area  is  one-fourth  that  of 
the  square  or  circle  formed  with 
cd.  In  the  scale  of  solids,  with 
the  mark  at  4,  ab  will  be  the  dia- 
meter of  a  sphere  or  edge  of  a 
cube  whose  solidity  is  one-fourth 
that  of  a  sphere  or  cube  whose  dia- 
meter or  edge  is  cd. 

The  graduation  of  these  instru- 
ments cannot  always  be  relied 
upon,  and  the  accuracy  of  the  pro- 
portions  is  also  affected  by  any 
change  in  the  length  of  the  legs, 
occasioned  by  use  or  breaking. 
It  would  be  well  to  test  their  accuracy  before  purchasing. 

7.  Beam  compasses.  These  are  for  describing  larger  arcs 
than  would  be  possible  with  the  other  compasses.  There  are 
different  forms  of  these.  In  all  of  them  there  is  a  beam  of 
wood,  or  metal,  with  two  collars,  one  carrying  a  pen  or  pencil 
point,  and  the  other  a  needle  point.  Fig.  6  shows  a  conve- 
nient form  of  this  instrument.  In  this  the  tops  of  the  collars 
are  left  open  so  that  they  can  be  used  on  any  straight  edge, 


DRAWING   INSTRUMENTS   AND   MATERIALS.  5 

being  damped  to  it  by  means  of  screws.  The  needle  point  is 
clamped  at  one  end,  while  the  other  collar  can  be  clamped  at 
any  distance  from  it.  By  turning  the  screw  a  the  point  b  can 
be  moved  slightly,  thus  enabling  one  to  adjust  nicely  the 


distance  between  5  and  c.  In  some  forms  there  is  a  scale 
upon  the  beam,  but  ihis  is  not  necessary  and  makes  it  more 
expensive. 

8.  Drawing  pen.  Fig.  7.  This  is  made  of  two  flat  pointed 
steel  blades,  one  of  which  has  a  hinge-joint  at  its  base,  thus 
allowing  the  blades  to  be  sufficiently  separated  for  cleaning 


or  sharpening.  The  distance  between  the  points  is  regulated 
by  means  of  a  screw  connecting  the  two  blades.  This  distance 
determines  the  width  of  the  lines.  Do  not  select  one  in  which 
the  blades  are  much  curved. 

9.  Drawing  board.  This  is  an  indispensable  part  of  the 
draughtsman's  outfit.  It  should  be  made  of  a  thoroughly 
seasoned  light  wood;  pine  is  generally  used.  A  board 
twenty -eight  inches  long,  twenty  inches  wide,  and  about  one 
inch  thick  is  a  convenient  size.  Each  end  should  be  finished 


INDUSTRIAL   DRAWING. 


with  a  cleat,  as  shown  in  Fig.  8,  to  prevent  warping.    Each 
side  should  be  rubbed  to  a  smooth  plane  surface,  with  fine 


sand-paper.    The  edges  should  be  straight  and  at  right  angles 
to  each  other. 

10.  J  square.    This  consists  of  a  long  thin  blade,  with  a 
head  at  right  angles  to  it ;  fig.  9  is  a  good  pattern,  where  the 


blade  is  simply  laid  upon  the  head  and  screwed  to  it.    Any 
fine-grained  hard  wood  will  do  for  material,  pear-wood  is  very 


DRAWING  INSTRUMENTS   AND   MATERIALS.  7 

good.  Sometimes  metal  is  used,  and  for  accuracy  it  would 
be  the  best ;  but  as  it  rusts  and  tarnishes  easily,  the  paper  is 
liable  to  be  soiled.  The  blade  should  be  about  the  same 
length  as  the  board.  For  a  blade  thirty  inches  in  length,  in 
order  to  be  sufficiently  stiff,  the  width  should  be  at  least  two 
and  a  half  inches,  and  the  thickness  one-eighth  of  an  inch. 
The  head  is  generally  made  of  one  piece,  although  sometimes 
of  two,  one  on  either  side  of  the  blade,  as  shown  in  fig.  10. 
One  of  these  pieces  is  fixed  while  the  other  can  be  clamped 
at  any  angle  with  the  blade ;  so  that,  if  the  square  is  turned 
over  and  used  with  this  side  of  the  head  against  the  edge  of 
the  board,  parallel  lines  may  be  drawn  making  a  correspond- 
ing angle  with  the  sides  of  the  board.  This  is  not  used  very 
much,  as  it  cannot  be  clamped  sufficiently  tight  to  prevent 
displacement  by  a  slight  rap,  or  any  great  pressure  upon  the 
blade. 

11.  Triangles.  These  are  made  of  different  materials,  as 
wood,  metal,  rubber,  etc.,  the  last  being  the  best ;  metal  tri- 
angles are  open  to  the  same  objection  that  metal  squares  are. 
Triangles  are  made  solid  or  open,  as  shown  in  fig.  11,  the 


open  ones  being  the  most  convenient  to  handle.  The 
draughtsman  requires  two  of  these,  one  having  angles  of  90°, 
45°,  45°,  while  the  other  has  angles  of  90°,  60°,  30°.  A  con- 
venient size  is  from  six  to  ten  inches  on  side. 

12.  Irregular  curves.  These  are  for  tracing  curves,  other 
than  arcs  of  circles,  which  are  determined  by  points.  Those 
made  of  thin  pear-wood  will  be  found  to  answer  well.  They 
have  a  great  variety  of  shapes,  so  that  one  is  at  a  loss  which 
to  select.  Choose  one  with  curves  of  both  small  and  large 
radius. 


8  INDUSTRIAL  DBAWING. 

13.  India  ink.    It  is  important  to  have  this  material  of  the 
best  quality,  as  it  is  used  exclusively  for  all  black  lines. 
Good  India  ink,  when  broken  across,  should  have  a  shining 
and  somewhat  golden  lustre.    If  an  end  of  it  is  wetted  and 
rubbed  on  the  thumb-nail,  it  should  have  a  pasty  feel,  free 
from  grains,  and  exhale  an  odor  of  musk ;  when  dried  the 
end  should  present  a  shiny  and  golden-hued  surface.     Ink  of 
inferior  quality  is  of  a  dull  bluish  color,  and  when  wetted 
and  rubbed  on  the  nail  feels  granular ;  also  when  rubbed  up 
in  water  it  settles ;  whereas  good  ink  remains  thoroughly  dif- 
fused through  the  water. 

14.  Colors.    Windsor  and  Newton's  water  colors  are  con- 
sidered the  best.     There  are  two  sizes  of  cakes,  called  the 
whole  and  half  cakes.     The  moist  colors  which  come  in  porce- 
lain dishes  are  equally  good,  and  are  preferred  by  some,  as 
the  cakes  are  liable  to  crack  and  crumble. 

15.  Ink  saucers.     These  come  in  nests  of  four  or  six,  and 
are  very  convenient  for  preserving  the  colors  after  they  are 
prepared.     One  forms  a  cover  for  another  and  thus  keeps 
out  the  dust,  besides  keeping  the  colors  moist  for  some  time. 

16.  Pencils.     These  should  be  of   the  hardest  and  best 
quality.     A.  W.  Faber's  hexagonal  pencils  are  the  best.     HH 
and  HHH 9xv  good  numbers  for  mechanical  drawing. 

17.  Brushes.     The  best  brushes  are  made  of  sable  hair,  of 
which  there  are  two  varieties,  the  red  and  black.     The  red  is 
somewhat  the  cheapest,  but  is  about  as  good  as  the  black. 
Camel's-hair  brushes  will  answer  very  well,  if  one  does  not 
want  to  afford  the  sable.     In  selecting  brushes  see  that  they 
come  to  a  point  when  moistened  with  water  ;  they  should 
keep  this  point  when  used. 

18.  India  rubber.     Fine  vulcanized  rubber  is  the  best  for 
removing  pencil  marks.     Sponge  rubber  is  good  for  cleaning 
drawings. 

19.  Thumb  tacks.     These  are  for  fastening  the  paper  to 
the  drawing  board,  when  it  is  not  necessary  to  stretch  it. 
The  best  have  the  steel  point  riveted  into  the  head.     The 
top  of  the  head  should  be  slightly  rounded,  the  edges  being 
thin,  so  as  to  allow  the  J  square  to  pass  over  readily. 

20.  Horn  centre.     This  is  used  in  case  many  circles  are  to 


DRAWING   INSTRUMENTS   AND   MATERIALS. 


9 


.  by  17 
"  20 
"  22 
"  24 
"  27 
"  30 

Elephant  

.      23  in 

.  by  28 
"  34 
"  34 
"  40 
"  53 

Columbier 

23   ' 

Atlas,  
Double  Elephant  .  .  . 
Antiquarian. 

...  26   ' 

...27   * 
...31   ' 

Emperor  .  .  . 

..  48   ' 

be  drawn  from  the  same  centre.  The  needle  point  rests  upon 
this  instead  of  the  paper,  to  prevent  the  wearing  of  a  large 
hole.  It  is  made  of  a  piece  of  transparent  horn,  with  three 
little  steel  points  to  prevent  slipping. 

21.  Drawing  paper. — Watmau's  drawing  paper  is  consid- 
ered the  best.     It  comes  in  sheets  of  standard  sizes,  as  follows : 

Cap 13  in.  by  17 

Demy 15 

Medium 17 

Eoyal 19 

Super  Royal 19 

Imperial 22 

There  are  two  kinds  of  this  paper,  the  rough-  or  cold-pressed, 
and  the  smooth-  or  hot-pressed ;  the  smooth  paper  is  better 
for  finished  line  drawings,  while  the  rough  surface  takes  color 
better ;  does  not  show  erasures,  and  is  better  for  general  work. 
On  holding  the  paper  up  to  the  light  the  maker's  name  is  seen 
in  water  lines,  and  when  it  can  be  read,  the  nearest  surface  is 
considered  the  right  side.  If  there  is  any  difference  between 
the  two,  it  is  supposed  to  be  in  favor  of  this  side.  The  double 
elephant  sheet  is  of  a  good  weight  and  size  for  general  work. 
For  smaller  sizes  cut  it  in  halves  or  quarters. 

22.  Tracing  paper.     This  is  a  thin  paper  prepared  so  as  to 
be  sufficiently  transparent  to  allow  the  lines  of  a  drawing  to 
show  through,  for  the  purpose  of  copying.     It  comes  in  sheets 
or  by  the  roll.     The  best  paper  is  tough,  transparent,  and 
without  any  greasiness. 

23.  Tracing  cloth.     This  is  for  the  same  purpose  as  tracing 
paper.     It  is  better  than  paper  for  preserving  copies,  or  for 
those  that  are  to  be  subjected  to  any  wear,  as  in  the  case  of 
working  drawings.     It  comes  in  rolls  of  twenty-four  yards, 
and  of  different  widths.     That  which  has  a  dull  surface  on 
one  side  is  better  for  pencil  marks  or  for  tinting. 


24.  As  it  would  be  impossible  in  most  oases  to  make  a 
drawing  the  same  size  as  the  object,  it  is  r  ecessary  to  resort 
to  the  use  of  scales  in  order  to  preserve  the  relative  positions 


10  INDU8TBIAL  DBA  WING. 

of  the  lines,  and  also  to  be  able  to  determine  what  proportion 
the  drawing  bears  to  the  object.  These  scales  consist  of  dis- 
tances of  different  lengths,  divided  with  great  accuracy. 
Various  materials  are  used  in  their  construction,  such  as 
ivory,  wood,  metal,  paper,  etc.  Ivory  is  the  best,  but  quite 
expensive,  while  those  made  of  boxwood  are  good,  and  are 
generally  used.  Metal  is  expensive  and  tarnishes  easily, 
while  paper  soon  wears  out. 

It  is  well  in  making  a  drawing  to  use  as  large  a  scale  as 
convenient,  so  that  the  small  parts  of  the  object  may  be 
brought  out  distinctly  and  accurately. 

The  scale  of  a  drawing  should  always  be  given  upon  it,  and 
in  the  case  of  a  working  drawing,  the  measurements  also. 
By  giving  the  measurements  the  time  of  the  workman  is  saved, 
since  he  follows  the  dimensions  rather  than  the  drawing,  in 
case  they  differ. 

The  dimensions  are  given  in  figures,  using  one  accent  to  in- 
dicate feet  and  two  accents  for  inches,  as  2',  read  two  feet,  and 
3",  three  inches.  Where  feet  and  inches  are  to  be  written,  the 
figures  are  separated  by  a  comma  simply,  as  6',  4".  Where 
only  inches  are  to  be  written,  use  a  cipher  in  place  of  feet,  as 
0',  6".  This  is  not  always  necessary,  but  upon  drawings  where 
many  of  the  dimensions  are  given  in  feet  and  fractions  of  a  foot, 
it  is  well  to  use  the  cipher,  as  it  may  save  mistakes  in  reading. 

To  indicate  the  points  between  which  the  measurement 
reads,  small  arrow  heads  are  used  at  the  points;  as  seen  upon 
PI.  XVI.  Fig.  144.  When  the  arrow  heads  are  some  distance 
apart,  they  are  connected  by  a  fine  dotted  line.  This  line  may 
be  broken  at  the  centre  to  allow  the  measurements  to  be  in- 
serted, or  the  line  may  be  continuous,  and  the  figures  be  placed 
just  above  it  or  on  it.  To  make  the  measurements  more 
noticeable,  red  ink  is  generally  used  for  the  figures  and  arrow 
heads,  and  blue  ink  for  the  dotted  lines.  All  the  horizontal 
measurements  should  be  written  from  left  to  right,  the  verti- 
cal being  written  from  the  bottom  to  the  top. 

25.  A  scale  of  equal  parts.  (PL  I.  Fig.  7.)  This  is  a  small 
flat  instrument  of  ivory.  On  both  sides  of  this  instrument 
we  shall  find  a  number  of  lines  drawn  lengthwise,  and 
divided  crosswise  by  short  lines,  against  which  numbers  are 


DRAWING   INSTRUMENTS.  1] 

written.  It  is  not  our  object,  in  this  place,  to  give  a  full 
description  of  this  scale,  but  simply  of  that  portion  which  is 
termed  the  scale  of  equal  parts.  A  scale  of  equal  parts  con- 
sists of  a  number  of  inches,  or  of  any  fractional  part  of  an 
inch,  set  off  along  a  line ;  the  first  inch,  or  fractional  part  to 
the  left,  being  subdivided  into  ten  or  twelve  equal  portions. 
When  the  divisions  of  the  line  are  inches,  the  scale  is  some- 
times termed  an  inch  scale;  if  the  divisions  are  each  three 
quarters  of  an  inch,  it  is  termed  a  three  quarter  inch  scale;  and 
so  on,  for  any  other  fractional  part  of  an  inch.  These  different 
scales  are  usually  placed  on  the  same  side  of  the  ivory  scale ; 
the  inch  scale  being  at  the  bottom,  the  three  quarter  inch 
scale  next  above ;  next  the  half  inch  scale,  and  so  on.  The 
inch  scale  is  usually  marked  IN.  on  the  left ;  the  three  quarter 
inch  scale  with  the  fraction  f ;  the  others,  in  like  manner, 
with  the  fractious  $,  *,  and  so  on.  The  first  division  of  each 
of  these  scales  is  usually  subdivided,  along  the  bottom  line 
of  the  scale,  into  ten  equal  parts,  by  short  lines;  and  just 
above  these  short  lines  will  be  found  others,  somewhat 
shorter,  which  divide  the  same  division  into  twelve  equal 
parts.  It  is  not  usual  to  place  any  number  against  the  first 
division ;  against  the  short  line  which  marks  the  second 
division,  the  number  1  is  written;  at  the  third  division 
the  number  2 ;  and  so  on  to  the  right. 

26.  Manner  of  using  the  scale.  Geometrical  drawings  of  ob- 
jects are  usually  made  on  a  smaller  scale  than  the  real  size  of  the 
object.  In  making  the  geometrical  drawing  of  a  house,  for 
example,  the  drawing  would  have  to  be  very  much  smaller 
than  the  house,  in  order  that  a  sheet  of  paper  of  moderate 
dimensions  may  contain  it.  We  must  then,  in  this  case,  a& 
in  all  others,  select  a  suitable  scale  for  the  object  to  be 
represented.  Let  us  suppose  that  the  scale  chosen  for  the 
example  in  question  is  the  inch  scale  ;  and  that  each  inch  on 
the  drawing  shall  correspond  to  ten  feet  on  the  house.  As  a 
foot  contains  twelve  inches,  it  is  plain,  that  an  inch  on  the 
drawing  will  correspond  to  120  inches  on  the  house ;  thus 
any  line  on  the  drawing  will  be  the  one  hundred  and 
twentieth  part  of  the  corresponding  line  on  the  house ;  and 
were  we  to  make  a  model  of  the  house,  on  the  same  scale  aa 


12  INDUSTRIAL   DRAWING. 

the  drawing,  the  height  or  the  breadth  of  the  model  would 
be  exactly  the  one  hundred  and  twentieth  part  of  the  height 
or  the  breadth  of  the  house.  By  bearing  these  remarks  in 
mind,  the  manner  of  using  this,  or  any  other  scale,  will 
appear  clearer.  Suppose,  for  example,  we  wish  to  take  off 
from  the  scale,  by  a  pair  of  dividers,  the  distance  of  twenty- 
three  feet ;  we  observe,  in  the  first  place,  that  as  each  inch 
corresponds  to  ten  feet,  two  inches  will  correspond  to  twenty 
feet ;  and  as  each  tenth  of  an  inch  corresponds  to  one  foot, 
three  tenths  will  correspond  to  three  feet.  We  then  place 
one  point  of  the  dividers  at  the  division  marked  2,  and 
extend  the  other  point  to  the  left  through  the  two  divisions 
of  an  inch  each,  and  thence  along  the  third  division  so  as 
to  take  in  three  tenths.  From  the  scale,  here  described,  we 
cannot  take  off  any  fractional  part  of  a  foot,  with  accuracy. 
If  we  had  wished  to  have  taken  off  twenty-three  feet  and  a 
half,  for  example,  we  might  have  extended  the  point  of  the 
dividers  to  the  middle  between  the  division  of  three  tenths 
and  four  tenths ;  but,  as  this  middle  point  is  not  marked  on 
the  scale,  the  accuracy  of  the  operation  wil]  depend  upon  the 
skill  with  which  we  can  judge  of  distances  by  the  eye.  To 
avoid  any  error,  from  want  of  skill,  we  must  resort  to  one  of 
the  scales  of  the  fractional  parts  of  an  inch.  Taking  the 
half  inch  scale,  for  example,  its  principal  divisions  being 
halves  of  an  inch,  will,  therefore,  correspond  to  five  feet ;  the 
first  of  its  principal  divisions  being  subdivided,  like  the  one 
on  the  inch  scale,  one  tenth  of  it  will  correspond  to  half  a 
foot,  or  six  inches.  So  that  from  this  scale  we  can  take  off 
any  number  of  feet  and  half  feet.  The  quarter  inch  scale 
being  the  half  of  the  half  inch  scale,  its  small  subdivisions 
will  correspond  to  a  quarter  of  a  foot,  or  to  three  inches.  The 
small  subdivisions  on  the  eighth  of  an  inch  scale,  in  like 
manner,  correspond  to  one  inch  and  a  half.  We  thus  observe 
that,  by  suitably  using  one  of  the  above  scales,  we  can  take 
off  any  of  the  fractional  parts  of  a  foot  on  the  inch  scale,  cor- 
responding to  a  half,  a  quarter,  or  an  eighth. 

Let  us  now  suppose,  as  a  second  example  in  the  use  of  the 
inch  scale,  that  we  had  to  make  a  drawing  of  a  house  on  a 
scale  of  one  inch  to  twelve  feet.  We  observe,  in  the  first 


DRAWING  INSTRUMENTS.  IB 

place,  as  twelve  feet  contain  144  inches,  and  as  one  inch,  ou 
the  drawing,  corresponds  to  144  inches  on  the  house  in  its 
true  size,  every  line  on  the  drawing  will  be  the  one  hundred 
and  forty-fourth  part  of  the  corresponding  lines  on  the  house. 
In  this  case,  instead  of  using  the  division  of  the  inch  sub- 
divided into  tenths,  we  use  the  divisions  of  twelfths,  which 
are  just  above  the  tenths ;  each  twelfth  corresponding  to  one 
foot.  We  next  observe^  in  using  the  twelfths,  what  we  found 
in  using  the  tenths ;  that  is,  we  cannot  obtain,  from  the  inch 
scale,  any  part  corresponding  to  a  fractional  part  of  a  foot ; 
but,  taking  the  half  inch  scale,  we  find  the  first  half  inch 
subdivided  in  the  same  way  as  the  inch  scale  into  twelfths ; 
and,  therefore,  each  twelfth  on  the  half  inch  scale  will  corres- 
pond to  half  a  foot,  or  six  inches.  In  like  manner  the 
twelfths  on  the  quarter  inch  scale  will  correspond  to  three 
inches ;  and  the  twelfths  on  the  eighth  of  an  inch  scale  to  one 
inch  and  a  half. 

27.  Diagonal  scale  of  equal  parts.  (PI.  I.  Fig.  8.)  In  the 
two  examples  of  the  use  of  an  ordinary  scale  of  equal  parts,  we 
found,  that  the  smallest  fractional  part  of  a  foot,  that  any  of 
the  scales  examined  would  give,  was  one  inch  and  a  half. 
But,  as  all  measurements  of  objects  not  larger  than  a  house 
are  commonly  taken  either  in  feet  and  inches,  or  in  feet  and 
the  decimal  divisions  of  a  foot,  it  is  very  desirable  to  have  a 
scale  from  which  we  can  obtain  either  the  tenths  of  a  foot,  if 
the  decimal  numeration  is  used ;  or  the  inches,  or  twelfths, 
when  the  duodecimal  numeration  is  used.  To  effect  these 
purposes,  a  scale,  called  a  diagonal  scale  of  equal  parts,  has 
been  imagined;  and  this  scale  is  frequently  found  on  the 
small  ivory  scale,  on  which  the  other  scales  of  equal  parts  are 
marked.  A  diagonal  scale  of  equal  parts  may  be  either  an 
inch  scale,  or  a  scale  of  any  fractional  part  of  an  inch.  The 
first  inch,  or  division,  is  divided  into  ten,  or  twelve  equal 
parts ;  the  other  inches  to  the  right  are  numbered  like  th& 
ordinary  scales.  Below  the  top  line,  on  which  the  inches  are 
marked  off,  we  find  either  ten,  or  twelve  lines,  drawn  parallel 
to  the  top  line,  and  at  equal  distances  from  each  other.  We 
next  observe  perpendicular  lines,  drawn  from  the  top  line, 
acaoss  the  parallel  lines,  to  the  bottom  line,  at  the  points  OD 


14  INDUSTRIAL  DRAWING, 

the  top  line  which  mark  the  divisions  of  inches.  But,  from 
the  subdivisions  on  the  top  line,  we  find  oblique  lines  drawn 
to  the  bottom  line ;  these  oblique  lines  being  also  parallel  to 
each  other,  and  at  equal  distances  apart.  The  direction  of 
these  oblique  lines,  termed  diagonal  lines,  and  from  which 
the  scale  takes  its  name,  is  now  to  be  carefully  noticed,  in 
order  to  obtain  a  right  understanding  of  the  use  of  the  scale. 
By  examining,  in  the  first  place,  the  bottom  line  of  the  scale, 
we  observe  that  it  is  divided,  in  all  respects,  precisely  like 
the  top  line.  In  the  next  place,  we  observe,  that  the  first 
diagonal  line  to  the  left,  10,  is  drawn  from  the  end  of  the  top 
line,  to  the  point  mt  which  marks  the  first  subdivision  to  the 
left  on  the  bottom  line.  The  other  diagonal  lines,  we  observe, 
are  drawn  parallel  to  the  first  one,  on  the  left,  and  from  the 
points  of  the  subdivisions  on  the  top  line ;  the  last  diagonal 
line,  on  the  right,  being  drawn  from  the  last  subdivision  on 
the  top  line,  to  the  end  of  the  first  division  of  the  bottom 
line.  On  examining  the  perpendicular  lines,  we  find,  that 
they  divide  all  the  parallel  lines,  from  top  to  bottom,  into 
equal  parts  of  an  inch  each.  We  find,  also,  that  all  the  sub- 
divisions, on  the  same  parallel  lines,  made  by  the  diagonal 
lines,  are  equal  between  the  diagonal  lines,  each  being  equal 
to  the  top  subdivisions.  But  when  we  examine  the  small 
divisions,  on  the  parallel  lines,  contained  between  the  first 
perpendicular  line,  10 — w,  on  the  left,  and  the  diagonal  line 
next  to  it,  we  find  these  small  divisions  increase  in  length 
from  the  top  to  the  bottom.  In  like  manner  we  find  that 
the  small  divisions,  on  the  parallel  lines,  between  the  dia- 
gonal line,  1 — r,  on  the  right,  and  the  perpendicular,  0 — r, 
next  to  it,  decrease  in  length  from  the  top  to  bottom.  Now,  it 
is  by  means  of  these  decreasing  subdivisions  that  we  get  the 
tenths,  or  twelfths  of  one  of  the  top  subdivisions,  according 
as  we  find  ten  or  twelve  parallel  lines  below  the  top  line. 

Let  us  suppose,  for  example,  that  we  have  ten  parallel 
lines  below  the  top  line ;  then  the  small  subdivision,  on  the 
first  parallel  line,  below  the  top  line,  contained  between  the 
diagonal  line,  1 — r,  on  the  right,  and  the  perpendicular, 
0 — r,  would  be  nine  tenths  of  the  top  subdivision ;  that  ia 
nine  tenths  of  a  foot,  if  the  toj>  subdivision  corresponds  to 


DRAWING  INSTRUMENTS.  15 

one  foot.  The  next  small  division  below  this  would  be  eight 
tenths;  the  next  below  seven  tenths;  and  so  on  down 
Now,  if  we  take  the  small  subdivisions,  between  the  per- 
pendicular line,  on  the  left,  and  the  first  diagonal  line, 
10 — m,  the  one  below  the  top  line  will  be  one  tenth  of  the 
top  subdivision ;  the  next  below  this  two  tenths ;  and  so  on, 
increasing  by  one  tenth,  as  we  proceed  towards  the  bottom. 

Let  us  suppose,  for  an  example  in  the  use  of  this  scale,  that 
the  drawing  is  made  to  a  scale  of  one  inch  to  ten  feet ;  and 
that  we  wish  to  take  off  from  the  scale  sixteen  feet  and  seven 
tenths.  With  the  dividers  slightly  open  in  one  hand,  we  run 
the  point  of  the  dividers  down  along  the  perpendicular, 
marked  0 — r,  next  to  the  diagonal  line  on  the  right,  until  we 
come  to  the  parallel  line  on  which  the  small  division  seven 
tenths  is  found ;  this  will  be  the  third  line  below  the  top  line. 
We  place  the  point  of  the  dividers  on  this  parallel  line,  at  a, 
where  the  perpendicular  line,  numbered  0 — r,  crosses  it,  and  we 
extend  the  other  point  to  the  right,  to  b,  to  the  perpendicular 
marked  1,  along  the  parallel  line,  to  take  in  one  inch,  or  ten 
feet.  We  then  keep  the  right  point  of  the  dividers  fixed,  at 
b ;  and  extend  the  other  point  to  the  left,  to  c,  taking  in  the 
small  division  seven  tenths,  and  six  of  the  subdivisions  along 
the  parallel  line.  The  length,  thus  taken  in  between  the  two 
points  of  the  dividers,  will  be  the  required  distance  on  the 
scale. 

From  the  description  of  the  diagonal  scale,  and  the  example 
of  its  use  just  explained,  the  arrangement  of,  and  the  manner 
of  using  any  other  diagonal  scale  will  be  easily  learned.  We 
have  first  to  count  the  number  of  parallel  lines  below  the  top 
line,  and  this  will  show  us  the  smallest  fractional  part  of  the 
equal  subdivisions  of  the  top  line  that  we  can  obtain,  by 
means  of  the  scale.  If,  for  example,  we  should  find  one 
hundred  of  these  lines  below  the  top  line,  then  we  could 
obtain  from  the  scale,  as  small  a  fraction  of  the  top  sub- 
divisions as  one  hundredth  ;  and  counting  downwards  on  the 
left,  or  upwards  on  the  right,  the  short  lines,  between  the 
extreme  diagonal  lines  and  the  perpendicular  lines  next  tr 
them,  we  could  obtain  from  the  one  hundredth  to  ninety 
nine  hundredths  of  the  top  subdivisions. 


1C 


INDUSTRIAL  DRAWING. 


Remarks.  In  the  preceding  examples  we  have  taken  the 
scales  as  corresponding  to  feet,  because  dimensions  of  objects 
of  ordinary  magnitude  are  usually  expressed  in  feet,.  When 
the  drawing  represents  objects  of  considerable  extent,  we 
then  should  use  a  scale  of  an  inch  to  so  many  yards,  or 
miles;  as  in  the  case  of- a  map  representing  a  field,  or  an 
entire  country.  When  the  drawing  represents  objects  whose 
principal  dimensions  are  less  than  a  foot  we  then  use  a  scale 
of  an  inch  to  so  many  inches.  In  some  cases  where  we  wish 
to  represent  very  minute  objects,  which  cannot  be  drawn 
accurately  of  their  natural  size,  we  use  a  magnified  scale,  that 
is  a  scale  which  gives  the  dimensions  on  the  drawing  a 
certain  number  of  times  greater  than  those  of  the  object 
represented.  For  example,  in  a  drawing  made  to  a  scale  of 
three  inches  to  one  inch,  the  dimensions  on  the  drawing 
would  be  three  times  those  of  the  corresponding  lines  on  the 
object. 

28.  A  convenient  form  of  diagonal  scale  for  obtaining 
twelfths  of  a  foot,  is  shown  in  Fig.  12,  where  ab,  be,  represent 
6  b 


/\5 

/       \* 

/               \3 

/                      \2 

/                             \/ 

/                                    \ 

Pro.  12. 

feet,  while  the  horizontal  distances  of  the  points  1,  2,  3,  etc., 
from  the  vertical  line  through  b  are  each  equal  to  a  cor- 
responding number  of  inches. 

29.  The  scales  represented  in  Figs.  7  and  8,  PI.  I,  although 

1  used  to  some  extent,  are  rather  short  and 
-«ss@mMMM^B!»~  not  as  convenient  to  use  as  those  where  the 

2  graduation  runs  to  the  edge,  so  that  by  lay- 
jng  the  scale  on  the  paper  the  required  dis- 
tance  can  be  marked  off  directly  without 
using  the  dividers.     The  frequent  use  of  the 
dividers  in  taking  distances  from  the  scales, 
defaces  the  lines  of  the  scale  and  should  be 
avoided.     In  Fig.  13  are  shown  sections  of 

no.  is.  scales  in  general  use.    These  are  all  made 


DBA  WING   INSTRUMENTS   AND  MATERIALS.  17 

with  thin  edges,  and  with  the  graduation  running  to  the  very 
edge.  On  Nos.  1  and  2  there  would  be  one,  or  possibly  two 
scales  for  each  edge,  while  on  No.  3  there  can  be  at  least  six 
scales  in  all,  two  to  each  edge,  one  on  either  side. 

30.  Fig.  14  gives  a  side  view  of  Nos.  1  or  2.     There  are 
four  scales  upon  this,  two  upon  each  edge,  one  being  double 


1  "M  1  "M  1   '  1 


the  other.  The  scales  are  duodeciraally  divided,  the  divisions 
representing  feet  and  inches.  The  following  scales  are  the 
most  useful :  •§-,  J,  £,  1,  J-,  f ,  1^,  3.  Two  flat  ones,  or  one  tri- 
angular, will  contain  all  of  these. 

31.  Fig.  15  represents  what  is  called  a  chain  scale.     Such 
scales  are  numbered  10,  20,  30,  40,  50,  60,  according  to  the 


i.  i  1  I  M 


number  of  divisions  contained  in  an  inch,  and  are  the  scales 
in  common  use.  The  upper  scale  in  Fig.  15  is  numbered  20, 
that  is,  an  inch  is  divided  into  20  equal  parts  and  might  be 
used  in  practice  for  a  scale  of  two  or  twenty  chains  to  the 
inch ;  two  or  twenty  feet  to  the  inch,  or  two  or  twenty  miles 
to  the  inch.  One  triangular,  or  three  flat  scales,  will  contain 
all  of  those  mentioned  above.  The  duodecimally  divided 
scales  are  most  useful  for  mechanical  and  geometrical  draw- 
ing, while  the  chain  scales,  as  their  name  indicates,  are  used 
mostly  for  plotting  surveys. 

32.  After  the  drawing  of  an  object  has  been  completed,  it 
is  necessary  to  state  upon  it  the  ratio  between  the  lines  of  the 
drawing  and  those  of  the  object.  This  is  often  done  by  the 
use  of  a  fraction,  as  a  scale  of  •£,  f,  i,  etc.  It  is  understood 
in  such  cases  that  the  numerator  refers  to  the  drawing  and 
the  denominator  to  the  object,  and  that  the  numerator  is  equal 
2 


18  INDUSTRIAL   DRAWING. 

to  the  denominator ;  for  example  in  the  scale  \  we  should 
understand  that  one  inch  on  the  drawing  equals  four  inches 
on  the  object.  In  the  scale  £  1  in.  =  8  in.  or  1£"  =  1  foot. 
Instead  of  always  expressing  scales  by  the  use  of  fractions,  it 
is  often  stated  what  the  scale  is  to  a  foot,  as  for  example, 
"  Scale  £  in.  =  1  foot,"  this  expressed  fractionally  would  be 
^f.  Scale  £  in.  =  1  foot  would  be  the  same  as  T^-. 

33.  Protractor.  This  is  the  instrument  in  most  common 
use  for  laying  off  angles.  It  consists  of  a  semicircle  of  thin 
metal  or  horn  (Fig.  16),  the  circumference  of  which  is  divided 


into  180  equal  parts,  termed  degrees,  and  numbered,  both 
ways,  from  0°  to  180°,  by  numbers  placed  at  every  tenth  divi- 
sion. As  the  protractor  is  only  divided  into  degrees,  subdivi- 
sions of  a  degree  can  be  marked  off  by  means  of  it  only  by 
judging  by  the  eye.  A  little  practice  will  enable  one  to  set 
off  a  half,  or  even  a  quarter  of  a  degree  with  considerable 
accuracy,  when  the  circumference  is  not  very  small.  Where 
greater  accuracy  is  required,  protractors  of  a  more  compli- 
cated form  are  used.  Some  of  the  ivory  scales  are  graduated 
so  that  they  can  be  used  as  protractors,  but  they  cannot  be 
depended  upon  for  accurate  work.  Horn  protractors  are 
convenient  to  use  as  they  are  transparent,  but  they  are  liable 
to  warp.  Tracing  paper  has  recently  been  used  to  print  pro- 
tractors upon.  Although  not  very  durable,  they  have  this 
advantage  that  they  may  be  made  of  larger  sizes,  some  being 
fourteen  inches  in  diameter ;  being  upon  tracing  paper,  they 
are  transparent,  and  also  cheap. 

There  are  other  forms  of  scales  more  or  less  complex,  and 
seldom  used,  which  it  is  not  necessary  to  describe  here. 


USB  AND  CAEE   OF   INSTRUMENTS.  19 


CHAPTER  II. 

USE  AND   CAEE   OF   INSTRUMENTS. 

1.  Good  tools  are  not  only  necessary  for  good  work,  but 
they  should  be  kept  in  perfect  order.     This  is  particularly 
true  in  drawing.     The  draughtsman  should  keep  at  hand  a 
clean  old  linen  rag,  and  a  piece  of  soft  washleather,  to  be 
used  for  cleaning  and  wiping  the  instruments  before  and 
after  using.     The  pen  should  be  wiped  dry  before  laying  it 
aside,  and  the  soil  from  perspiration  cleaned  with  the  wash- 
leather  in  like  manner.     With  proper  care  a  set  of  good  in- 
struments will  last  a  lifetime. 

2.  Compasses.     In  describing  circles  keep  both  legs  nearly 
vertical.     This  can  be  accomplished  by  means  of  the  joint  in 
each  leg.     By  keeping  the  needle  in  a  vertical  position,  it  will 
not  make  a  larger  hole  than  necessary  in  the  paper.     Usually 
one  hand  is  sufficient  when  describing  circles ;  but  when  the 
lengthening  bar  is  used,  it  is  better  to  steady  the  needle  point 
with  one  hand,  while  the  circle  is  described  with  the  other. 
When  inking  a  circle,  complete  it  before  taking  the  pen  from 
the  paper,  because  without  great  care  the  place  where  the 
lines  join  will  show. 

3.  Dividers.     These   are  used  for  transferring  distances 
from  one  part  of  a  drawing  to  another,  or  from  scales  to  the 
drawing.     One  hand  is  sufficient  for  their  use ;  but  care  must 
be  taken  when  transferring  distances  not  to  change  the  posi- 
tion of  the  points  by  clumsy  handling.     A  little  care  and 
practice  will  enable  one  to  manage  them  easily.     When  tak- 
ing any  distance  with  the  dividers,  open  the  points  wider  than 
necessary,  and  then  bring  to  the  required  distance  by  pressing 
them  together.     In  case  it  is  required  to  lay  off  the  same  dis- 
tance a  number  of  times  on  a  straight  line,  it  is  better  to  step 
the  distance  off,  keeping  one  of  the  points  on  the  line  all  the 


20  INDUSTBIAL   DRAWING. 


time.  In  doing  this,  do  not  turn  the  instrument  continuously 
in  one  direction,  but  reverse  at  each  step,  so  that  the  moving 
point  will  pass  alternately  to  the  right  and  left  of  the  line. 
A  single  trial  will  convince  one,  by  the  ease  with  which  the 
instrument  can  be  handled,  of  the  advantage  of  the  last 
method.  Care  should  be  taken  when  using  the  dividers  not 
to  injure  the  surface  of  the  paper.  When  laying  off  distances, 
do  not  make  holes  in  the  paper ;  a  very  slight  prick  is  suffi- 
cient, and  this  can  afterwards  be  marked  with  the  pencil. 

4.  Drawing  pen.    This  is  one  of  the  most  important  instru- 
ments, and  should  never  be  of  inferior  quality.     It  is  used 
for  inking  the  lines  of  a  drawing  after  it  has  been  pencilled. 
The  ink  can  be  placed  between  the  blades  by  means  of  a 
small  brush  or  a  strip  of  paper ;  but  a  more  convenient  way 
is  to  dip  the  edges  of  the  pen  into  the  ink,  and  they  will 
generally  take  up  enough ;  in  case  they  do  not,  breathe  upon 
them,  and  then  they  will  readily  take  the  ink.     After  taking 
the  ink  in  this  way,  always  wipe  the  outside  of  the  blades. 
When  inking,  incline  the  pen  slightly  to  the  right  in  the 
direction  of  the  line,  taking  care  that  both  points  touch  the 
paper.     Do  not  press  too  hard  against  the  ruler,  nor  try  to 
keep  the  point  of  the  pen  too  near  the  ruler.     If  the  line  is 
to  be  heavy,  or  broad,  the  pen  should  be  moved  along  rather 
slowly,  otherwise  the  edges  of  the  line  will  appear  rough. 
Always  draw  the  lines  from  left  to  right.     In  case  the  ink 
does  not  flow  readily  from  the  pen,  try  it  upon  a  piece  of 
waste  paper,  or  outside  the  border  line ;  a  piece  of  blotting 
paper  is  also  good  for  this  purpose.     If  these  means  fail,  pass 
between  the  blades  the  corner  of  a  piece  of  firm  paper. 
When  the  ink  gets  too  thick,  it  dries  rapidly  in  the  pen  and 
occasions  considerable  annoyance  ;  in  such  a  case,  add  a  little 
water  to  the  ink,  and  mix  well  before  using. 

5.  Sharpening  pen.     As  the  pen  is  used  a  great  deal,  the 
points  are  worn  away,  changing  their  form  as  well  as  thick- 
ness, making  it  impossible  to  do  nice  work.     Every  draughts- 
man should  be  able  to  sharpen  his  pen  and  keep  it  in  good 
order ;  with  a  little  practice  and  attention  to  the  following 
directions,  one  will  soon  be  able  to  sharpen  a  pen.     Having 
obtained  a  fine  grained  oilstone,  first  screw  the  points  together, 


USE   AND   CARE   OF   INSTRUMENTS.  21 

then  draw  the  pen  over  the  stone,  keeping  it  in  a  perpendicu- 
lar plane,  and  turning  it  at  the  same  time,  so  that  the  curve 
at  the  point  shall  take  the  shape  shown  in  N\  1,  Fig.  17. 


This  is  a  better  shape  than  either  N"os.  2  or  3,  although  3 
might  do  for  heavy  lines ;  but  for  all  lines,  1  is  the  best 
shape.  It  is  well,  however,  to  have  two  pens ;  one  being  kept 
sharp  for  fine  lines,  while  the  other  is  used  for  heavy  lines, 
and  need  not  be  sharpened  as  often.  Having  obtained  the 
right  shape,  separate  the  points  a  little  by  means  of  the  screw, 
and  then  lay  each  blade  in  turn  upon  the  stone — keeping  the 
pen  at  an  angle  of  about  30°  with  the  face  of  the  stone — and 
grind  off  the  thickness  so  as  to  bring  to  an  edge  along  the 
point  of  this  curve.  For  fine  lines  this  edge  should  be  pretty 
thin  ;  not  a  knife  edge,  which  would  cut  the  paper.  See  that 
the  edges  of  both  blades  are  of  the  same  thickness,  and  the 
points  of  the  same  length ;  then  take  the  screw  out  and  open- 
ing the  blades,  apply  the  inside  of  each  to  the  stone  just 
enough  to  take  off  any  edge  that  may  have  been  turned  over. 
Next,  screw  up  the  blades,  and  try  the  points  on  the  thumb 
nail  to  see  if  the  edges  are  too  sharp  ;  if  they  cut  the  nail,  the 
sharp  edge  should  be  taken  off  by  drawing  the  pen  very  lightly 
over  the  stone  as  in  the  first  step.  Next,  try  the  pen  with  ink 
on  paper,  and  the  character  of  the  line  made  will  show  whether 
anything  more  is  needed. 

6.  Drawing  board.  The  sides  should  be  plane  surfaces, 
and  their  accuracy  may  be  tested  by  placing  the  edge  of  a 
ruler  across  the  board  in  several  positions,  and  observing 
whether  it  coincides  throughout,  in  every  position,  with  the 
surface.  The  edges  of  the  board  should  be  perfectly  straight, 
and  if  one  cannot  detect  any  inaccuracy  by  the  eye  alone, 
they  can  be  tested  by  applying  a  straight  edge.  It  is  not 
essential,  however,  that  the  edges  should  be  exactly  at  right 
angles  to  each  other,  provided  the  head  of  the  square  is  used 
only  on  one  edge  for  any  given  drawing. 


22 


INDUSTRIAL   DRAWING. 


7.  T  square.    By  placing  the  head  against  any  edge  of  the 
board,  and  moving  it  along,  at  the  same  time  drawing  lines 
along  the  edge  of  the  blade  in  its  successive  positions,  we  have 
a  series  of  parallel  lines.     If  now  the  head  is  placed  against 
either  adjoining  edge,  and  lines  are  drawn  as  before,  we  shall 
have  two  sets  of  parallel  lines  at  right  angles  to  each  other  ; 
that  is,  if  the  two  edges  of  the  board  are  exactly  at  right 
angles,  and  also  the  blade  of  the  square  exactly  at  right  angles 
to  the  head.    Now,  as  it  is  impossible  to  be  sure  that  these 
angles  are  correct  at  all  times,  and  as  the  lines  drawn  while 
the  head  is  moved  along  any  edge  would  be  parallel,  whether 
the  blade  be  at  right  angles  to  the  head  or  not,  it  is  better  to 
use  the  head  upon  only  one  edge,  and  depend  upon  the  trian- 
gles for  perpendicular  lines.     It  is  customary  and  more  con- 
venient to  use  the  head  upon  the  left-hand  edge  of  the  board, 
controlling  its  movements  with  the  left  hand,  while  the  right 
is  free  to  draw  the  lines.     In  moving  from  one  position  to 
another,  take  hold  of  the  head  instead  of  the  blade.     Always 
see  that  the  head  is  against  the  edge  before  drawing  a  line, 
and  use  the  upper  edge  in  ruling. 

8.  Triangles.     As  there  are  two  triangles,  having  the  angles 
90°,  45°,  45°,  and  90°,  60°,  30°,  respectively,  it  is  evident  that, 
by  placing  either  of  these  triangles  against  the  blade  of  the 
square,  lines  may  be  drawn  making  corresponding  angles. 
As  before  suggested,  this  is  the  best  way  to  draw  perpendicu- 
lars.    In  case  parallel  lines  are  to  be  drawn,  making  angles 
different  from  those  of  the  triangles,  it  may  be  possible  to 


accomplish  it  by  a  combination  of  both  triangles  with  the 
square ;  but  it  is  difficult  to  hold  so  many  pieces  in  place,  se 


USE   AND   CAEE   OF  INSTRUMENTS.  23 

that  it  is  better  to  use  either  one  with  the  square,  moving  the 
blade  so  that  the  edge  of  the  triangle  shall  coincide  with  a 
line  already  drawn  at  the  desired  angle ;  then,  by  holding  the 
blade  in  place,  and  moving  the  triangle,  lines  may  be  drawn 
parallel  to  the  first.  To  test  the  right  angle  of  the  triangle, 
place  it  against  the  square  as  in  Fig.  18,  and  draw  the  line 
a  b}  then  turn  the  triangle  over,  as  in  the  dotted  position,  and 
draw  another  line  along  the  vertical  edge,  taking  care  that  the 
two  lines  start  from  the  same  point  a.  Now,  if  these  lines 
correspond  throughout,  the  angle  at  the  base  is  a  right  augle ; 
if  there  is  any  deviation,  it  will  indicate  what  change  should 
be  made  to  correct  it. 

Maniier  of  using  the  triangles  for  drawing  lines  which  are 
to  be  either  parallel  or  perpendicular  to  another  line. 

Let  A  B  (PI.  I.  Fig.  4)  be  a  line  to  which  it  is  required  to 
draw  parallel  lines  which  shall  respectively  pass  through  the 
points,  (7,  D,  E,  etc.,  on  either  side  of  A  B. 

1st.  Place  the  longest  side,  a  5,  of  the  triangle  so  as  to 
coincide  accurately  with  the  given  line ;  and,  if  its  other  sides 
are  unequal,  taking  care  to  have  the  next  longest  of  the  two, 
a  c,  towards  the  left  hand.  2d.  Keeping  the  triangle  accu- 
rately in  this  position,  with  the  left  hand,  place  the  edge  of 
the  ruler  against  the  side,  a  <?,  and  secure  it  also  in  its  position 
with  the  left  hand.  3d.  Having  the  instruments  in  this 
position,  slide  the  triangle  along  the  edge  of  the  ruler  towards 
one  of  the  given  points,  as  G  for  example,  until  the  side,  a  J, 
is  brought  so  near  the  point  that  a  line  drawn  with  the  pencil 
along  a  b  will  pass  accurately  through  the  point.  4th.  Keep 
the  triangle  firmly  in  this  position,  with  one  of  the  fingers 
of  the  left  hand,  and  with  the  right  draw  the  pencil  line 
through  the  point.  Proceed  in  the  same  way  with  respect  to 
the  other  points. 

Remarks.  "When  practicable,  the  first  position  of  the 
triangle  should  be  so  chosen  that  the  side,  a  b,  can  be  brought 
in  succession  to  pass  through  the  different  points  without 
having  to  change  the  position  of  the  ruler. 

After  drawing  each  line  the  triangle  should  be  brought 
back  to  its  first  position,  in  order  to  detect  any  error  from  an 


24  INDUSTRIAL  DRAWING. 

accidental  change  in  the  position  of  the  ruler  during  the 
operation;  and  it  will  generally  be  found  not  only  most 
convenient  but  an  aid  to  accuracy  to  draw  the  line  through 
the  highest  point  first,  and  so  on  downwards ;  as  the  eye  will 
the  more  readily  detect  any  inaccuracy,  in  comparing  the 
positions  of  the  lines  as  they  are  successively  drawn.  It  will 
also  be  found  most  convenient  to  place  the  triangle  first 
against  the  ruler  and  adjust  them  together,  in  the  first  position 
along  A  J3. 

It  will  readily  appear  that  either  side  of  the  triangle  may 
be  placed  against  the  ruler,  or  the  given  line.  The  draftsman 
will  be  guided  on  this  point  by  the  positions  and  lengths  of 
the  required  parallels. 

Let  A  B  (PL  I.  Fig.  5)  be  a  line  to  which  it  is  required  to 
draw  a  perpendicular  at  a  point,  6",  upon  the  line,  or  through 
a  point,  I},  exterior  to  it. 

1st.  Place  the  longest  side,  a.  b,  of  the  triangle  against  the 
given  line,  and  directly  beneath  the  given  point  (7,  or  D,  with 
the  ruler  against  a  c,  as  in  the  last  case. 

2d.  Confining  the  ruler  firmly  with  the  left  hand,  shift  the 
position  of  the  triangle  so  as  to  bring  the  other  shorter  side, 
b  c,  against  the  ruler. 

3d.  Slide  the  triangle  along  the  ruler,  until  the  longest  side, 
a  b,  is  brought  upon  the  given  point  6r,  or  I)  ;  and,  confining 
it  in  this  position,  draw  with  the  pencil  a  line  through  (7,  or 
D.  The  line  so  drawn  is  the  required  perpendicular. 

Another  method  of  drawing  a  perpendicular  in  like  cases 
is  to  place  the  ruler  against  the  given  line,  and,  holding  it 
firmly  in  this  position,  to  place  one  of  the  shorter  sides  of  the 
triangle  against  the  edge  of  the  ruler,  and  then  slide  the  tri- 
angle along  this  edge  until  the  other  shorter  side  is  brought 
upon  the  given  point,  when,  the  triangle  being  confined  in 
this  position,  the  required  perpendicular  can  be  drawn.  But 
this  is  usually  neither  so  accurate  nor  so  convenient  a  method 
as  the  preceding  one. 

Remark.  The  accuracy  of  the  two  methods  just  described 
will  depend  upon  the  accuracy  of  the  triangle.  If  its  right 
angle  is  not  perfectly  accurate,  the  required  perpendicular 
will  not  be  true.  This  method  also  should  only  be  resorted 


USE   AND   CAEE   OF   INSTRUMENTS.  25 

to  for  perpendiculars  of  short  lengths.  In  other  cases  one  of 
the  methods  to  be  described  farther  on  should  be  used  as  be- 
ing less  liable  to  error. 

9.  Irregular  curves.    Having  determined  the  points  through 
which  the  curve  is  to  pass,  the  edge  of  the  instrument  is  shifted 
about  until  it  is  brought  in  a  position  to  coincide  with  three 
or  more  consecutive  points  «,  d,  c  (PL  I.  Fig.  11)  of  the  curve 
x-y  to  be  traced.    When  the  position  chosen  satisfies  the  eye, 
that  portion  of  the  curve  is  traced  in  pencil  which  will  pass 
through  the  points  coinciding  with  the  edge  of  the  instrument. 
The  instrument  is  again  shifted  so  as  to  bring  several  points, 
contiguous  to  the  one  or  other  end  of  the  portion  of  the  curve 
just  traced,  to  coincide,  as  before,  with  the  edge  of  the  instru- 
ment ;  taking  care  that  the  new  portion  of  the  curve  shall 
form  a  continuation  of  the  portion  already  traced.     The  ope- 
ration is  -continued  in  this  way  until  the  entire  curve  is  traced 
in  pencil ;  it  is  then  put  in  ink  by  going  over  the  pencil  line 
with  the  pen,  using  the  instrument  as  a  guide,  as  at  first. 

The  successful  use  of  this  instrument  demands  some  atten- 
tion and  skill  on  the  part  of  the  draughtsman,  in  judging  by 
the  eye  the  direction  of  the  curve  from  the  position  of  its 
points.  When  thus  employed,  curves  of  the  most  complicated 
forms  can  be  traced  with  the  greatest  accuracy. 

When  similar  curves  are  to  be  drawn  on  both  sides  of  a 
line — as  in  an  ellipse — mark  on  the  edge  of  the  instrument 
the  limits  of  the  part  used  in  drawing  the  curve  on  one  side, 
and  then  turn  it  over  and  draw  the  opposite  side  between  the 
same  limits. 

10.  Preparation  ofiiik.     Place  a  few  drops  of  water  in 
the  saucer  and  rub  the  ink  in  it  until  the  liquid  is  pretty  thick 
and  black.    After  a  little  experience  one  can  tell  by  the  ap- 
pearance when  it  has  arrived  at  the  proper  consistency  for 
use ;  it  is  not  well,  however,  to  trust  to  sight  alone,  but  also 
to  test  it  by  making  a  wide  line  with  the  drawing  pen  upon 
a  piece  of  paper,  to  see  if  it  is  perfectly  black  when  dry.     In 
testing  in  this  way,  go  over  the  line  only  once.     The  cake 
should  always  be  wiped  dry  after  using  to  prevent  cracking. 
Keep  the  saucer  covered  when  not  filling  the  pen.     Ink  may 
be  kept  moist  for  some  time  in  a  covered  saucer ;  but  if  it 


26  INDUSTRIAL  DRAWING. 

should  happen  to  dry,  it  is  better  to  wash  it  all  out  and  start 
anew. 

11.  Pencils.    For  drawing  straight  lines  the  pencil  shouLl 
not  be  cut  to  a  round  point,  but  to  a  flat  thin  edge  -like  a 
wedge ;  the  flat  side  being  laid  against  the  edge  of  the  ruler 
when  drawing  lines.     It  is  well  to  have  another  pencil  sharp- 
ened to  a  round  point  for  making  dots  and  marking  points. 
For  the  compasses  the  pencil  point  should  be  round. 

When  the  pencil  becomes  dull,  or  makes  a  heavy  line,  it 
may  be  sharpened  by  rubbing  on  any  hard  surface,  as  a  piece 
of  rough  paper,  a  piece  of  fine  sand  paper,  or  a  fine  file. 
Long  pencils  are  more  convenient  to  handle ;  when  they  get 
short,  trim  them  down  for  the  compasses. 

12.  Stretching  paper.    When  it  is  desired  to  make  a  small 
line  drawing,  requiring  no  tinting,  nor  much  time  for  its  com- 
pletion, it  is  sufiicient  to  fasten  the  paper  to  the  board  with 
thumb-tacks;  but  for  large  drawings,  and  especially  those 
requiring  any  tinting  or  shading,  it  is  necessary  to  stretch 
the  paper  by  wetting,  and  then  fasten  to  the  board  with 
mucilage. 

To  do  this,  place  the  sheet  of  paper  upon  the  board,  right 
side  up,  and  turn  up  each  edge  of  the  paper  against  a  ruler 
placed  upon  it  about  half  an  inch  from  the  edge.  Then  take 
a  perfectly  clean  sponge,  and  with  clean  water  moisten  the 
upper  surface,  omitting  the  edges  that  are  turned  up;  in 
doing  this,  do  not  rub  the  paper,  but  merely  press  the  sponge 
upon  all  parts.  Let  it  stand  with  a  little  water  upon  the  sur- 
face for  about  fifteen  minutes — less  time  might  do,  but  that 
will  always  ensure  a  good  stretch — after  that  sop  up  the 
standing  water,  and  then  fasten  each  edge  of  the  paper  to  the 
board  with  thick  mucilage.  After  fastening  one  edge,  it  is 
better  to  fasten  the  opposite  edge  next,  instead  of  either  ad- 
joining edge.  Rub  the  edges  down  with  the  thumb  to  secure 
a  firm  joint.  Leave  the  board  in  a  horizontal  position  while 
the  paper  is  drying,  taking  care  not  to  place  near  the  fire. 

Some  prefer  not  to  wet  that  side  of  the  paper  upon  which 
the  drawing  is  to  be  made.  There  is  no  objection  to  it,  if  one 
is  only  careful  not  to  injure  the  surface  by  rubbing,  and  tc 
use  clean  water ;  besides,  it  is  a  little  more  convenient,  foj 


USE  AND  OAEE  OF  INSTEUMENT8.  27 

the  edges  are  not  likely  to  get  wet,  as  they  are  turned  up ;  it 
also  saves  the  trouble  of  turning  the  paper  over,  during  which 
operation  the  board  is  likely  to  get  wet,  and  the  mucilage  will 
not  be  so  likely  to  hold. 

When  stretching  a  large  sheet  where  the  centre  is  apt  to 
dry  before  the  edges,  so  that  there  is  danger  of  their  being 
pulled  up,  lay  a  damp  cloth  upon  the  centre,  away  from  the 
edges,  and  let  it  remain  until  they  become  dry. 

13.  Pencilling  a  drawing.     This  consists  in  locating  the 
lines  of  the  drawing  properly  and  accurately  upon  the  paper. 
In  doing  this,  do  not  make  any  unnecessary  lines,  or  any  lines 
unnecessarily  long ;  a  little  attention  given  to  this  will  make 
a  much  neater  drawing,  and  there  will  be  less  liability  to 
make  mistakes  when  inking.     The  lines  should  be  fine  and 
merely  dark  enough  to  be  seen.     As  a  rule  it  is  better  to  com- 
plete the  drawing  in  pencil  before  inking ;  to  this  rule,  how- 
ever, there  are  exceptions;  as  in  complicated  drawings  of 
machinery  where  it  is  well  to  ink  some  parts  before  finishing 
the  pencilling  of  the  whole. 

14.  Inking  a  drawing.     It  is  customary  in  inking  a  draw- 
ing to  ink  all  the  circles  and  arcs  before  the  right  lines,  as  it 
is  easier  to  make  the  lines  meet  the  arcs  than  the  arcs  the 
lines.     When  there  are  a  number  of  small  concentric  circles 
to  be  inked,  commence  with  the  smallest. 

After  inking  all  the  circles,  ink  the  horizontal  lines  next, 
commencing  at  the  top  and  working  down ;  then  ink  the  ver- 
tical lines,  commencing  at  the  left  and  going  to  the  right. 
Where  there  are  to  be  lines  of  different  widths,  ink  those  of 
the  same  width  before  changing  the  pen.  By  having  a  little 
system  one  will  work  more  rapidly. 

When  inking  a  complicated  drawing  before  or  after  it  is 
entirely  pencilled,  ink  first  those  parts  that  are  in  front,  and 
work  back,  thus  avoiding  the  danger  of  making  lines  full  that 
should  be  dotted. 

To  obtain  the  best  effect  in  a  line  drawing,  the  ink  should 
be  perfectly  black,  and  the  lines  smooth  and  even ;  the  thick- 
ness of  the  fine  and  heavy  lines  should  be  proportioned  to 
each  other  and  to  the  scale  of  the  drawing.  The  shade  lines 
should  also  be  correctly  located.  There  is  another  thing  that 


28  INDUSTRIAL   DRAWING. 

adds  much  to  the  looks  of  a  drawing,  and  that  is  perfect  inter- 
sections of  lines,  as  in  the  case  of  two  lines  meeting  at  a  point; 
neither  line  should  stop  before  reaching  the  point,  nor  run 
beyond,  as  is  more  likely  to  occur. 

If  the  drawing  is  to  be  shaded,  do  not  use  black  ink  for  the 
outlines,  as  light  ink  will  blend  into  the  shading  better ;  it  is 
well,  too,  to  make  the  lines  fine. 

It  is  sometimes  found  necessary  to  ink  a  straight  line  so 
that  it  shall  taper.  This  is  easily  accomplished  by  moving 
one  end  of  the  ruler  slightly  back,  the  other  end  remaining 
fixed,  and  going  over  the  line  again.  Where  a  number  of 
lines  meet  at  a  point,  care  must  be  taken  in  inking,  not  to  make 
a  large  blot  at  the  intersection ;  it  is  better  to  draw  the  lines 
from  rather  than  towards  the  point ;  wait  for  each  line  to  dry 
before  drawing  the  next.  Where  there  are  a  great  many 
lines,  the  angle  between  them  being  small,  they  will  unavoid- 
ably meet,  before  reaching  the  point ;  so  that  in  inking  it  is 
not  well  to  start  each  line  from  the  point,  but  only  from 
where  it  would  meet  the  last  line.  Let  about  every  third  line 
start  from  the  point. 

15.  Accuracy.     In  both  the  construction  of  a  drawing  in 
pencil  and  the  inking,  accuracy  should  be  the  aim  of  the 
draughtsman.     No  matter  how  simple  the  drawing,  let  the 
construction  be  exact.     The  importance  of  this  is  apt  to  be 
lost  sight  of  in  the  class  room  and  upon  elementary  work, 
but  will  be  manifest  when  one  comes  to  the  construction  of 
drawings  for  use.     Begin  right,  then,  and  cultivate  accuracy 
as  a  habit. 

16.  Cleaning  a  drawing.     To  remove  pencil  lines  a  piece 
of  vulcanized  India  rubber  is  necessary ;  but  to  remove  dirt 
there  is  nothing  better  than  a  piece  of  stale  bread,  using  the 
inside  of  a  crust.     This  seems  to  remove  the  dirt  without 
affecting  the  ink  lines,  and  leaves  the  paper  with  a  fine  gloss. 
Do  not  rely,  however,  upon  the  possibility  of   cleaning  a 
drawing  when  finished  ;  never  allow  it  to  get  very  much  soiled. 
Always  dust  with  a  brush,  the  drawing  board,  paper,  and 
instruments,  before  commencing  work.     When  a  drawing  is 
to  occupy  much  time,  paste  a  piece  of  waste  paper  on  the 
edge  of  the  drawing  board,  to  fold  over  the  drawing  whet: 


USB   AND   CAKE   OF   INSTRUMENTS.  29 

not  at  work.     By  cutting  the  paper,  one  can  arrange  so  as  to 
cover  part  of  the  drawing  while  working  on  another  part. 

17.  Cutting  off  a  drawing.    To  remove  the  paper  from 
the  board  after  the  drawing  is  finished ;  place  the  ruler  so  as 
to  coincide  with  a  pencil  line,  drawn  just  inside  the  line  of 
the  mucilage,  and  then  run  the  knife  along  the  ruler. 

Before  cutting  off  a  drawing  see  that  it  is  perfectly  dry,  as 
it  will  wrinkle  if  part  of  the  sheet  is  damp.  "When  cutting 
take  the  edges  in  order  round  the  board ;  not  crossing  from 
one  edge  to  the  opposite,  and  let  each  cut  commence  from  the 
last  one. 

18.  Tracing.    It  is  often  necessary  to  have  one  or  more 
copies  of  a  drawing.    By  the  use  of  tracing  paper  or  cloth, 
these  can  be  taken  from  the  drawing  already  made,  thus  sav- 
ing much  time.     The  paper  or  cloth  should  be  large  enough 
to  allow  the  tacks,  which  hold  it  down,  to  clear  the  drawing ; 
weights  may  be  used  in  case  it  is  not,  but  they  are  apt  to  be 
in  the  way.     Some  of  the  tracing  cloth  has  one  side  dull,  for 
the  use  of  a  pencil,  if  desired ;  let  the  tracing  be  made  upon 
the  glazed  side.     Tracing  cloth  will  take  color,  but  the  tint 
should  be  somewhat  thicker  for  this  purpose  than  for  ordinary 
tinting.     Apply  the  tint  to  the  dull  side  of  the  cloth ;  do  not 
try  to  tint  large  surfaces. 

19.  Accuracy  of  scales.     To  test  the  accuracy  of  the  gradu- 
ation, take  any  distance  with  the  dividers,  from  the  scale,  and 
compare  it  with  a  similar  distance,  at  different  parts  of  the 
scale.     It  is  well  to  test  the  graduation  near  the  ends,  as  it  is 
more  likely  to  be  inaccurate  there.     Another  way  of  testing 
is  by  placing  two  scales  together  to  see  if  they  correspond. 


PROBLEMS  OF  STRAIGHT  LINES.  31 


CHAPTER  III. 

CONSTRUCTION  OF  PROBLEMS  OF  POINTS  AND  STRAIGHT 

LINES. 

Prob.  1.  (PL  I.  Fig.  13.)  To  draw  a  straight  line  through 
two  given  points.  Let  A  and  B  be  two  given  points,  pricked 
into  the  surface  either  by  the  sharp  point  of  a  needle,  or  of  a 
lead  pencil. 

1st.  Make  with  the  lead  pencil  a  small  o,  or  round  thus  0, 
enclosing  each  point,  for  the  purpose  of  guiding  the  eye  in 
finding  the  point. 

2d.  Place  the  edge  of  the  ruler  in  such  a  position  near  the 
points  that  the  point  of  the  lead  pencil,  or  other  instrument 
used,  pressed  against  and  drawn  along  the  edge  of  the  ruler, 
will  pass  accurately  through  the  points. 

3d.  Place  the  pencil  upon,  or  a  little  beyond  the  point  J., 
on  the  left,  and  draw  it  steadily  along  the  edge  of  the  ruler 
until  its  point  is  brought  to  the  point  B,  on  the  right,  or  a 
short  distance  beyond  it. 

Remark.  To  draw  a  straight  line  accurately,  so  as  to  avoid 
any  breaks,  or  undulations  in  its  length,  and  have  its  breadth 
uniform,  demands  considerable  practice  and  skill.  The 
pressure  of  fingers  and  thumb  on  the  pencil  should  be  firm 
but  gentle,  as  well  as  the  pressure  of  the  pencil  against  the 
edge  of  the  ruler  and  upon  the  surface  of  the  drawing.  "We 
should  endeavor  to  get  into  the  habit  of  beginning  the  line 
exactly  at  the  one  point  and  finishing  it  with  the  same  pre- 
cision at  the  other ;  this  is  indispensable  in  the  case  of  ink 
lines,  but  although  not  so  important  in  pencil  lines,  as 
they  are  easily  effaced,  still  it  aids  in  giving  firmness  to  the 
hand  and  accuracy  to  the  eye  to  practise  it  in  all  cases.  The 


32  INDUSTRIAL  DRAWING. 

small  round  0  placed  about  the  dot  will  be  found  a  verj 
useful  adjunct  both  to  the  hand  and  eye  in  all  cases.  Draw- 
ing ink  lines  of  various  degrees  of  breadth  with  the  ordinary 
steel  or  quill  pen  will  be  also  found  excellent  practice  undei 
this  head. 

Lines  in  drawing  are  divided  into  several  classes  (PI.  I. 
Fig.  13  bis)  ws>full,  ~brolf.en,  dotted,  and  broken  and  dotted,  &c. ; 
these  again  are  divided  into  fine,  medium,  and  heavy,  according 
to  the  breadth  of  the  line.  A  fine  line  is  the  one  of  least  breadth 
that  can  be  distinctly  traced '  with  the  drawing  pen ;  the 
medium  line  is  twice  the  breadth  of  the  fine ;  and  the  heavy 
is  at  least  twice  the  breadth  of  the  medium. 

The  coarse  broken  line  consists  of  short  lines  of  about  fa 
of  an  inch  in  length,  with  blank  spaces  of  the  same  length 
between  them.  The  fine  broken  lines  and  spaces  are  fa  of 
an  inch. 

The  dotted  line  consists  of  small  elongated  dots  with  spaces 
of  the  same  between. 

The  broken  and  dotted  consists  of  short  lines  from  ^  to  \ 
of  an  inch  with  spaces  equal  in  length  to  the  lines  divided  by 
one,  two,  or  three  dots  at  equal  distances  from  each  other  and 
the  ends  of  the  lines. 

These  lines  may  also  be  fine,  medium,  or  heavy. 

When  a,  line  is  traced  with  quite  pale  ink  it  is  termed  a 
faint  line. 

The  lines  of  a  problem  which  are  either  given  or  are 
to  be  found  should  be  traced  in  full  lines  either  fine  or 
medium. 

The  lines  of  construction  should  be  broken  or  dotted. 

The  outlines  of  an  object  that  can  be  seen  by  a  spectator 
from  the  point  of  view  in  which  it  is  represented  should  be 
fall,  and  either  fine,  medium,  or  heavy,  according  to  the  par- 
ticular effect  that  the  draftsman  wishes  to  give.  The  portions 
of  the  outline  that  cannot  be  seen  from  the  assumed  point  of 
view,  but  which  are  requisite  to  give  a  complete  idea  of  the 
object,  should  be  dotted  or  broken. 

The  other  lines  are  used  for  conventional  purposes  by  the 
draftsman  to  show  the  connexion  between  the  parts  of  a 
problem,  &c.,  &c. 


PROBLEMS   OF  STRAIGHT  LINES.  33 

Prob.  2.  (PI.  II.  Fig.  14.)  To  set  of  a  given  distanje, 
along  a  straigJvt  line,  from  a  given  point  on  it. 

Let  C — D  be  the  line,  and  A  the  given  point. 

1st.  Mark  the  given  point  A,  as  in  the  preceding  pro- 
blem. 

2d.  Take  off  the  given  distance,  from  the  scale  of  equal 
parts,  with  the  dividers. 

3d.  Set  one  foot  of  the  dividers  on  A,  and  bring  the  other 
foot  upon  the  line,  and  mark  the  point  j?,  either  by  pricking 
the  surface  with  the  foot  of  the  dividers,  or  by  a  small  dot 
made  on  the  line  with  the  sharp  point  of  a  lead  pencil. 

When  the  distance  to  be  set  off  is  too  small  to  be  taken 
off  from  the  scale  with  accuracy,  proceed  as  follows : — 

1st.  Take  off  in  the  dividers  any  convenient  distance 
greater  than  the  given  distance,  and  set  it  off  from  A  to  b. 

2d.  Take  off  the  length  by  which  A  b  is  greater  than  the 
given  distance  and  set  it  off  from  b  to  c,  towards  A  ;  the  part 
A  c  will  be  the  required  distance. 

Remark.  A  given  distance,  as  the  length  of  a  line,  or  the 
distance  between  two  given  points,  is  sometimes  required  to 
be  set  off  along  some  given  line  of  a  drawing.  This  is  done 
by  a  series  of  operations  precisely  the  same  as  just  described. 
In  using  the  dividers,  they  must  be  held  without  stiffness, 
care  being  taken  not  to  alter  the  opening  given  to  them,  in 
taking  off  the  distances,  until  the  correctness  of  the  result  has 
been  carefully  verified,  by  going  over  the  operation  a  second 
time.  Particular  care  should  be  paid  to  the  manner  of  hold- 
ing the  dividers  in  pricking  points  with  them,  to  avoid 
changing  their  opening,  as  well  as  making  too  large  a  hole 
in  the  drawing  surface. 

Prob.  3.  (PI.  II.  Fig.  15.)  To  set  off,  along  a  straight  line, 
any  number  of  given  equal  distances. 

Let  G  D  be  the  straight  line,  and  the  given  number  of 
equal  distances  be  eight. 

1st.  Commence  by  marking  a  point  A,  on  the  line,  in  the 
usual  manner,  as  a  starting  point. 

2d.  The  number  of  equal  divisions  being  even,  take  off  in 
the  dividers  from  the  scale  their  sum,  and  set  it  off  from  A  tc 
8,  and  mark  the  point  8. 
3 


34  INDUSTRIAL   DRAWING. 

3d.  Take  from  the  scale  half  the  sum  total,  and  set  it  off 
from  A  to  4 ;  taking  care  to  ascertain  that  the  dividers 
will  accurately  extend  from  4  to  8,  before  marking  the 
point  4. 

4th.  Take  off  from  the  scale  the  fourth  of  the  sum  total, 
and  set  it  off  respectively  from  A  to  2,  and  4  to  6 ;  taking 
care  to  verify,  as  in  the  preceding  operation,  the  distances 
2—4,  and  6—8. 

5th.  Take  off  from  the  scale  the  given  equal  part,  and  set 
it  off  from  A  to  1 ;  from  2  to  3,  &c. ;  taking  care  to  verify 
the  distances  as  before. 

Remark.  When  the  number  of  equal  distances  is  odd, 
commence  by  setting  off  from  the  starting  point,  as  just 
described,  an  even  number  of  equal  distances,  either  greater 
or  smaller  than  the  given  odd  number  by  one,  taking  in  pre- 
ference the  even  number  which  with  its  parts  is  divisible  by 
two ;  if,  for  example,  the  odd  number  is  7,  then  take  8  as  the 
even  number  to  be  first  set  off;  if  it  is  5  then  take  4  as  the 
even  number.  Having  as  in  the  first  example,  set  off  8  parts 
we  take  only  the  seven  required  parts ;  and  in  the  second 
having  set  off  4  parts  only  we  add  on  the  remaining  fifth  part 
to  complete  the  required  whole. 

The  reason  for  using  the  operations  just  given  instead  of 
setting  off  each  equal  part  in  succession,  commencing  at  the 
starting  point,  is,  if  there  should  be  the  least  error  in  taking 
off  the  first  equal  part  from  the  scale,  this  error  will  increase 
in  proportion  to  the  total  number  of  equal  parts  set  off,  so 
that  the  whole  distance  will  be  so  much  the  longer  or  shorter 
than  it  ought  to  be,  by  the  length  of  the  error  in  the  first 
equal  distance  multiplied  by  the  number  of  times  it  has  been 


Prob.  4.  (PI.  II.  Fig.  16.)  From  a  point  on  a  right  line  to  set 
off  any  number  of  successive  unequal  distances. 

Let  C  D  be  the  given  line,  and  A  the  point  from  which  the 
first  distance  is  to  be  reckoned ;  and,  for  example,  let  the 
distances  be  respectively  A  b  equal  20  feet ;  b  c  equal  8 ;  c  d 
equal  15  feet ;  and  d  B  equal  25  feet. 

1st.  Commence  by  adding  into  one  sum  the  total  number 
of  distances,  which  in  this  case  is  68  feet. 


PROBLEMS   OF   STRAIGHT  LINES.  35 

2d.  Take  off  from  the  scale  of  equal  parts  this  total,  and 
set  it  off  from  A  to  B. 

3d.  Add  up  the  three  first  distances  of  which  the  total  is 
43  feet ;  take  this  off  from  the  scale,  and  set  it  off  from  A 
tod. 

4th.  Take  off  the  distance  d  B  in  the  dividers,  and  apply  it 
to  the  scale  to  verify  the  accuracy  of  the  construction. 

5th.  Set  off  successively  the  distances  A  b  equal  20  feet ; 
and  A  c  equal  28  feet ;  and  verify  by  the  scale  the  distances 
b  GJ  and  c  d. 

Remark  The  object  of  performing  the  operations  in  the 
manner  here  laid  down  is  to  avoid  carrying  forward  any 
inaccuracy  that  might  be  made  were  the  respective  distances 
set  off  separately.  The  verifications  will  serve  to  check,  as 
well  as  to  discover  any  error  that  may  have  been  made  in 
any  part  of  the  construction. 

Prob.  5.  (PL  II.  Fig.  17.)  To  divide  a  given  line,  or  the 
distances  between  two  given  points,  into  a  given  number  of  equal 


Let  A  B  be  the  distance  to  be  divided;  and  let,  for 
example,  the  number  of  its  equal  parts  be  four. 

Take  off  the  distance  A  B  in  the  dividers,  and  apply  it  to 
the  scale  of  equal  parts,  then  see  whether  the  number  of 
equal  parts  that  it  measures  on  the  scale  is  exactly  divisible 
by  4,  or  the  number  of  parts  into  which  A  B  is  to  be  divided. 
If  this  division  can  be  performed,  the  quotient  will  be  one  of 
the  required  equal  parts  of  A  B.  Having  found  the  length 
of  one  of  the  equal  parts  proceed  to  divide  A  B  precisely  in 
the  same  way  as  in  Prob.  3. 

If  AB  cannot  be  divided  in  this  way,  we  shall  be  obliged 
to  use  the  ruler  and  triangle,  in  addition  to  the  dividers  and 
scale  of  equal  parts,  to  perform  the  requisite  operations,  and 
proceed  as  follows : — 

1st.  Through  the  point  B,  draw  with  the  ruler  and  pencil 
a  straight  line,  which  extend  above  and  below  the  line  A  B, 
so  that  the  whole  length  shall  be  longer  than  the  longest  side 
of  the  triangle  used.  The  line  G  D  should  make  nearly  a 
right  angle  with  A  B. 

2d.  Take  off  from  the  scale  of  equal  parts  any  distance 


36  INDUSTRIAL  DRAWING. 

greater  than  A  B,  which  is  exactly  divisible  by  4,  or  the 
number  of  parts  into  which  A  B  is  to  be  divided. 

3d.  Place  one  foot  of  the  dividers  at  J.,  and  bring  the 
other  foot  upon  the  line  CD,  and  mark  this  second  point  4, 
in  the  usual  way. 

4th.  Draw  a  straight  line  through  A  and  4. 

5th.  Divide,  in  the  usual  way,  the  distance  A  4  into  its 
four  equal  parts  A  1;  &c.,  and  mark  the  points  1,  2,  3,  &c. 

6th.  With  the  ruler,  triangle,  and  pencil  draw  lines  parallel 
to  CD,  through  the  points  3,  2,  and  1 ;  and  mark  the  points 
df,  c,  and  6,  where  these  parallel  lines  cross  A  B.  The  distances 
A  b,  b  c,  c  d,  and  d  B  will  be  equal  to  each  other,  and  each  the 
one  fourth  of  A  B. 

Remarks.  The  distance  A  4  may  be  taken  any  length 
greater  than  A  B  the  line  to  be  divided.  It  will  generally  be 
found  most  convenient  to  take  a  length  over  twice  that  of 
AB. 

The  line  B  C  is  drawn  so  as  to  make  nearly  a  right  angle 
with  A  B,  in  order  that  the  points  where  the  lines  parallel  to 
it  cross  A  B  may  be  distinctly  marked.  Attention  to  the 
selection  of  lines  of  construction  is  of  importance,  as  the 
accuracy  of  the  solution  will  greatly  depend  on  this  choice. 
In  this  Prob.,  for  example,  the  line  CD  might  have  been 
taken  making  any  angle,  however  acute,  with  A  B,  without 
affecting  the  principle  of  the  solution  ;  but  the  practical  result 
might  have  been  very  far  from  accurate,  had  the  angle  been 
very  acutey  from  the  difficulty  of  ascertaining  with  accuracy, 
by  the  eye  alone,  the  exact  point  at  which  two  lines  intersect 
which  make  a  very  acute  angle  between  them,  such  as  the  lines 
drawn  from  the  points  1,  2,  3,  &c.,  parallel  to  CD.  would 
have  made  with  A  B  had  the  angle  between  it  and  CD  been 
very  acute.  The  same  remarks  apply  to  the  selection  of  arcs 
of  circles  by  which  points  are  to  be  found  as  in  figs.  18,  19, 
&c.  The  radii  in  such  cases  should  be  so  chosen  that  the 
arcs  will  not  intersect  in  a  very  acute  angle. 

Prob.  6.  (PI.  II.  Fig.  18.)  From  a  point  on  a  given  line  to 
construct  a  perpendicular  to  the  line. 

Let  CD  be  the  given  line,  and  A  the  point  at  which  it  is 
required  to  construct  the  perpendicular. 


PROBLEMS  OF  STRAIGHT  LINES.  37 

1st.  Having  fitted  the  pencil  point  to  the  dividers,  open  the 
legs  to  any  convenient  distance,  and  having  placed  the  stee] 
point  at  A ,  mark,  by  describing  a  small  arc  across  the  given 
line  with  the  pencil  point,  two  points  b  and  c,  on  either  side 
of  J.,  and  at  equal  distances  from  it. 

2d.  Place  the  steel  point  at  6,  and  open  the  legs  until  the 
pencil  point  is  brought  accurately  on  the  point  c  ;  then  from 
b,  with  the  distance  b  c,  describe  with  the  pencil  point  a  small 
arc  above  and  below  the  line,  and  as  nearly  as  the  eye  can 
judge  just  over  and  under  the  point  A. 

3d.  Preserving  carefully  the  same  opening  of  the  dividers, 
shift  the  steel  point  to  the  point  e,  and  describe  from  it  small 
arcs  above  and  below  the  line,  and  mark  with  care  the  points 
where  they  cross  the  two  described  from  b. 

4th.  With  the  ruler  and  pencil,  draw  a  line  through  the 
points  A  and  B,  extend  it  above  B  as  far  as  necessary  ;  this 
is  the  required  perpendicular. 

Remarks.  The  accuracy  of  the  preceding  construction  will 
depend  in  a  great  degree  upon  a  judicious  selection  of  the 
equal  distances  set  off  on  each  side  of  A  in  the  first  place ;  in 
the  opening  of  the  dividers  b  c  with  which  the  arcs  are 
described ;  and  upon  the  care  taken  in  handling  the  instru- 
ments and  marking  the  requisite  points. 

With  respect  to  the  two  equal  distances  A  b  and  A  c,  they 
may  be  taken  as  has  been  already  said  of  any  length  we 
please,  but  it  will  be  seen  that  the  longer  they  are  taken, 
provided  the  whole  distance  be  can  be  conveniently  taken  off 
with  the  dividers,  the  smaller  will  be  the  chances  of  error  in 
the  construction.  Because  the  greater  the  distance  b  c  the 
farther  will  the  point  J?,  where  the  two  arcs  cross,  be  placed 
from  J.,  and  any  error  therefore  that  may  happen  to  be  made, 
in  marking  the  point  of  crossing  of  the  arcs  at  B,  will  throw 
the  required  perpendicular  less  out  of  its  true  position  than 
if  the  same  error  had  been  made  nearer  to  the  point  A  ; 
moreover,  it  is  easier  to  draw  a  straight  line  accurately 
through  two  points  at  some  distance  apart  than  when  they 
are  near  each  other ;  particularly  if  the  line  is  required  to  be 
extended  beyond  either  or  both  of  the  points ;  for  if  any 
error  is  made  in  the  part  of  the  line  joining  the  two  points  it 


38  INDUSTRIAL  DRAWING. 

will  increase  the  more  the  farther  the  line  is  extended  eithei 
way  beyond  the  points. 

With  respect  to  the  distance  b  c,  with  which  the  arcs  are 
described,  this  might  have  been  taken  of  any  length  provided 
it  were  greater  than  A  b.  But  it  will  be  found  on  trial,  if  a 
distance  much  less  than  the  three  fourths  of,  or  much  greater 
than  b  c,  is  taken,  to  describe  the  arcs  with,  that  their  point 
of  crossing  cannot  be  marked  as  accurately  as  they  can  be 
when  the  distance  b  c  is  used. 

Attention  to  a  judicious  selection  of  distances,  &c.,  used  in 
making  a  construction,  where  they  can  be  taken  at  pleasure, 
is  of  great  importance  in  attaining  accuracy.  Where  points, 
like  JB,  are  to  be  found,  by  the  crossing  of  arcs,  or  of  straight 
lines,  we  should  endeavor  to  give  the  lines  such  a  position  that 
the  point  of  crossing  can  be  distinctly  made  out,  and  accu- 
rately marked;  and  this  will,  in  all  cases,  be  effected  by 
avoiding  to  place  the  lines  in  a  very  oblique  position  to  each 
other. 

A  further  point  to  secure  accuracy  of  construction  is  to 
obtain  means  of  proof,  or  verification.  In  the  construction 
just  made,  the  point  d  will  serve  as  a  means  of  verification ; 
for  the  perpendicular,  if  prolonged  below  A,  should  pass 
accurately  through  the  point  d  if  the  construction  is  correct. 

Prob.  7.  (PI.  II.  Fig.  19.)  From  a  point,  at  or  near  the 
extremity  of  a  given  line,  to  construct  a  perpendicular  to  the  line. 

Let  CD  be  the  line,  and  A  the  point. 

In  this  case  the  distance  A  C  being  too  short  to  use  it  as  in 
the  last  Prob.,  and  there  not  being  room  to  extend  the  line 
beyond  C,  a  different  process  must  be  used. 

1st.  Mark  any  point  as  a  above  CD,  and  between  A  and 
D. 

2d.  Place  the  foot  of  the  dividers  at  a,  and  open  the  legs 
until  the  pencil  point  is  brought  accurately  on  A ;  then 
describe  an  arc  to  cross  CD  at  £>,  and  produce  it  from  A  so 
far  above  it  that  a  straight  line  drawn  through  b  and  a  will 
cross  the  arc  above  A. 

3d.  Mark  the  point  b,  and  with  the  ruler  and  pencil  draw 
a  straight  line  through  a  and  b,  and  prolong  it  to  cross  the 
arc  at  B. 


PROBLEMS   OF   STRAIGHT  LINES.  39 

4th.  Mark  the  point  B  ;  and  with  the  ruler  and  pencil  draw 
a  line  through  A  and  B.  This  will  be  the  required  perpen 
dicular. 

Prdb.  8  (PI.  II.  Fig.  18.)  From  a  given  point,  above  or 
below  a  given  line,  to  draw  a  perpendicular  to  the  line. 

Let  CD  be  the  given  line,  and  B  the  given  point. 
1st.  Take  any  opening  of  the  dividers  with    the  pencil 
point,  and  placing  the  steel  point  at  B  describe  two  small  arcs, 
crossing  CDatb  and  c;  and  mark  carefully  these  points. 

2d.  Without  changing  the  opening  of  the  dividers,  place 
the  steel  points  successively  at  b  and  c,  from  which  describe 
two  arcs  below  CD,  and  mark  the  point  d  where  they  cross. 

3d.  With  the  pencil  and  ruler  draw  a  line  through  B  d. 
This  is  the  required  perpendicular. 

Remark.  The  distance  B  c,  taken  to  describe  the  first  arcs, 
should  as  nearly  as  the  eye  can  judge  be  equal  to  that  be 
between  them,  unless  the  given  point  is  very  near  the  given 
line. 

Verification.  If  the  construction  is  accurate,  the  distance 
A  b  will  be  found  equal  to  A  c  ;  and  A  B  equal  to  A  d. 

Prdb.  9.  (PI.  II.  Fig.  19.)  To  construct  the  perpendicular 
when  the  point  is  nearly  over  the  end  of  the  given  line. 

Let  B  be  the  given  point,  and  CD  the  given  line. 

1st.  Take  off  any  equal  number  of  equal  distances  from  the 
scale  with  the  dividers  and  pencil  point. 

2d.  Place  the  steel  point  at  B,  and,  with  the  distance 
taken  off,  describe  an  arc  to  cross  CD  at  b ;  and  mark  the 
point  b. 

3d.  Draw  a  line  through  B  b. 

4th.  Take  off  half  the  distance  B  6,  and  set  it  off  from  either 
B,  or  b  to  a,  and  mark  the  point  a. 

5th.  Place  the  steel  point  of  the  dividers  at  a,  and  stretch- 
ing the  pencil  point  to  b,  or  B,  describe  an  arc  to  cross  CD  at 
A,  and  mark  the  point  A. 

6th.  Draw  a  line  through  A  and  B.  This  is  the  required 
perpendicular. 

Verification.  Produce  B  A  below  CD,  and  set  off  A  d 
equal  to  A  B;  if  the  construction  is  accurate  b  d  will  be  found 
equal  to  Bb. 


4:0  INDUSTRIAL  DRAWING. 

Prob.  10.  (PI.  II  Figs.  18,  19.)  From  a  given  point  of  a 
line  to  set  off  a  point  at  a  given  distance  above  or  below  t\t, 
line. 

Let  A  be  the  given  point  on  the  line  CD. 

1st.  By  Prob.  6,  or  7,  according  to  the  position  of  A,  con- 
struct a  perpendicular  at  A  to  CD. 

2d.  Take  off  the  given  distance  and  set  it  off  from  A  along 
the  perpendicular,  according  as  the  point  is  required  above  or 
below  the  line. 

Remark.  If  the  point  may  be  set  off,  at  pleasure,  above  or 
below  CD,  we  may  either  construct  a  perpendicular  at  plea- 
sure, and  set  off  the  point  as  just  described,  or  we  may  take 
the  following  method,  which  is  more  convenient  and  expedi- 
tious, and,  with  a  little  practice,  will  be  found  as  accurate 
as  either  of  the  preceding. 

Take  off  the  given  distance  in  the  dividers.  Then  place 
one  foot  of  the  dividers  upon  the  paper,  and  describing  an 
arc  lightly  with  the  other,  notice  whether  it  just  touches, 
crosses,  or  does  not  reach  the  given  line.  If  it  crosses,  the 
position  taken  for  the  point  is  too  near  the  line,  and  the  foot 
of  the  dividers  must  be  shifted  farther  off;  if  the  arc  does  not 
ttieet  the  line  the  foot  of  the  dividers  must  be  brought  nearer 
to  the  line.  If  the  arc  just  touches  the  line  the  point  wheio 
the  stationary  foot  of  the  dividers  is  placed  being  marked 
will  be  a  point  at  the  required  distance  from  the  given 
line. 

Verification.  The  correctness  of  this  method  may  be 
verified  by  describing  from  a  point  set  off  by  either  of  the 
other  methods  an  arc  with  the  given  distance,  which  will  be 
found  just  to  touch  the  given  line. 

Prob.  11.  (PI.  II.  Fig.  20.)  Through  a  given  point  to  draw 
a  line  parallel  to  a  given  line. 

Let  A  be  the  given  point ;   CD  the  given  line. 

1st.  Place  one  foot  of  the  dividers  at  A,  and  bring  the 
other  foot  in  a  position  such  that  it  will  describe  an  arc  that 
shall  just  touch  CD  at  b. 

2d.  Without  changing  the  opening  of  the  dividers  place 
one  foot  at  a  point  J5,  near  the  other  end  of  CD,  so  that  the 
arc  described  with  this  opening  will  just  touch  the  line  at  c. 


PROBLEMS  OF  STRAIGHT  LINES.  41 

3d.  Having  marked  the  point  B,  draw  through  A  B  a  line. 
This  will  be  the  required  parallel. 

Verification.  Having  constructed  the  two  perpendiculars 
A  b,  and  B  c,  to  CD,  the  distance  b  c  will  be  found  equal 
to  A  B  if  the  construction  is  accurate. 

Prob.  12.  (PI.  II.  Fig.  20.)  To  draw  at  a  given  distanct 
from  a  given  line  a  parallel  to  the  line. 

Let  CD  be  the  given  line. 

1st.  Take  off  in  the  dividers  the  given  distance  at  which 
the  parallel  line  is  to  be  drawn. 

2d.  Find,  by  either  of  the  preceding  methods,  a  point  At 
near  one  end  of  CD,  and  a  point  B  near  the  other,  at  the 
given  distance  from  CD. 

3d.  Having  marked  these  points  draw  a  line  through  them. 
This  is  the  required  parallel. 

Verification.  The  same  proof  may  be  used  for  this  as  in 
Prob.  11. 

Prob.  13.  (PI.  II.  Fig.  21.)  To  transfer  an  angle  ;  or,  from  a 
point,  on  a  given  line,  to  draw  a  line  which  shall  make  with  the  given 
line  an  angle  equal  to  one  between  two  other  lines  on  the  drawing. 

Let  the  given  angle  to  be  transferred  be  the  one  bac 
between  the  lines  a  b  and  a  c.  Let  DJEbe  the  given  line,  and 
A  the  point,  at  which  a  line  is  to  be  so  drawn  as  to  make 
with  D  E  an  angle  at  A  equal  to  the  given  angle. 

1st.  With  the  dividers  and  pencil  point  describe,  from  o, 
with  any  opening,  an  arc,  and  mark  the  points  b  and  c,  where 
it  crosses  the  lines  containing  the  angle. 

2d.  Without  changing  this  opening,  shift  the  foot  of  the 
dividers  to  A,  and  describe  from  thence  an  arc  as  nearly  as 
the  eye  can  judge  somewhat  greater  than  the  one  6c,  and 
mark  the  point  B  where  it  crosses  D  E. 

3d.  Place  the  foot  of  the  dividers  at  6,  and  extend  the 
pencil  point  to  c. 

4th.  Shift  the  foot  of  the  dividers  to  B,  and,  with  the  same 
opening,  describe  a  small  arc  to  cross  the  first  at  C. 

5th.  Having  marked  the  point  C,  draw  a  line  through 
A  C.  This  line  will  make  with  D  E  the  required  angle. 

Prob.  14.  (PI.  II.  Fig.  22.)  From  a  point  of  a  given  line^ 
to  draw  a  line  making  an  angle  of  60°  with  the  given  line. 


42  INDUSTRIAL  DRAWING. 

Let  D  E  be  the  given  line,  and  A  the  given  point 

1st.  Take  any  distance  in  the  dividers  and  pencil  point,  and 
Bet  it  off  from  A  to  B. 

2d.  From  A  and  B,  with  the  same  opening,  describe  an 
arc,  and  mark  the  point  0  where  the  arcs  cross. 

3d.  Draw  a  line  through  A  C.  This  line  will  make  with 
the  given  one  the  required  angle  of  60°. 

Prob.  15.  (PI.  II.  Fig.  22.)  From  a  point  on  a  given  line  to 
draw  a  line  making  an  angle  0/"45°  with  it. 

Let  B  be  the  given  point  on  the  line  D  E. 

1st.  Set  off  any  distance  Ba,  along  D  E,  from  B. 

2d.  Construct  by  Prob.  6  a  perpendicular  to  D  E  at  a. 

3d.  Set  off  on  this  perpendicular  a  c  equal  to  a  B. 

4th.  Having  marked  the  point  c,  draw  through  Be  a  line. 
This  will  make  with  D  E  the  required  angle  of  45°. 

Prob.  16.  (PI.  II.  Fig.  23.)  To  divide  a  given  angle  into 
two  equal  parts. 

Let  B  A  0  be  the  given  angle. 

1st.  With  any  opening  of  the  dividers  and  pencil  point, 
describe  an  arc  from  the  point  A,  and  mark  the  points  b 
and  c,  where  it  crosses  the  sides  A  B  and  A  0  of  the 
angle. 

2d.  Without  changing  the  opening  of  the  dividers,  describe 
from  b  and  c  an  arc,  and  mark  the  point  D  where  the  area 
cross. 

3d.  Draw  a  line  through  A  D.  This  line  will  divide  the 
given  angle  into  equal  parts. 

Verification.  If  we  draw  a  line  through  b  c,  and  mark  the 
point  d  where  it  crosses  AD;  the  distance  b  d  will  be  found 
equal  to  c?  c,  and  the  line  b  c  perpendicular  to  A  D,  if  the 
construction  is  accurate. 

Remarks.  Should  it  be  found  that  the  point  of  crossing  at 
D  of  the  arcs  described  from  b  and  c  is  not  well  defined, 
owing  to  the  obliquity  of  the  arcs,  a  shorter  or  longer  dis- 
tance than  A  b  may  be  taken  with  which  to  describe  them, 
without  making  any  change  in  the  points  b  and  c  first  set 
off. 

Prob.  17.  (PI.  II.  Fig.  24.)  To  find  the  line  which  wH\ 
divide  into  two  equal  parts  the  angle  contained  between  two 


PROBLEMS   OF  CIRCLES,    &C.  43 

given  lines,  when  the  angular  point,  or  point  of  meeting  of  fhi 
two  lines,  is  not  on  the  drawing. 

Let  A  B  and  CD  be  the  two  given  lines. 

1st.  By  Prob.  10  set  off  a  point  at  b  at  any  distance  taken 
at  pleasure  from  A  B,  and  by  Prob.  11  draw  through  this 
point  a  line  parallel  to  A  B. 

2d.  Set  off  a  point  d  at  the  same  distance  from  D  0  as  b  is 
from  A  B,  and  draw  through  it  a  parallel  to  D  C ;  and  mark 
the  point  c  where  these  parallel  lines  cross. 

3d.  Divide  the  angle  bed  between  the  two  parallels  into 
two  equal  parts,  by  Prob.  16.  The  dividing  line  c  a  will  also 
divide  into  two  equal  parts  the  angle  between  the  given 
lines. 

Verification.  If  from  any  point,  as  o,  on  the  line  ca,  a 
perpendicular  o  m  be  drawn  to  A  B,  and  another  on  to  CD, 
these  two  perpendiculars  will  be  found  equal  if  the  construc- 
tion is  accurate. 


CONSTRUCTION  OP  PROBLEMS  OP  ARCS  OF  CIRCLES, 
STRAIGHT  LINES,  AND  POINTS. 

Prob.  18.  (PI.  II.  Fig.  25.)  Through  two  given  points  to 
describe  an  arc  of  a  circle  with  a  given  radius. 

Let  B  and  C  be  the  two  points. 

1st.  Take  off  the  given  distance  in  the  dividers  and  pencil 
point,  and  with  it  describe  an  arc  from  B  and  C  respectively, 
and  mark  the  point  A  where  the  arcs  cross. 

2d.  Without  changing  the  opening  of  the  dividers,  describe 
an  arc  from  the  point  A  through  B  and  C,  which  will  be  the 
one  required. 

Prob.  19.  (PI.  II.  Fig.  26.)  To  find  the  centre  of  a  circle,  or 
arc,  the  circumference  of  which  can  be  described  through  three 
given  points,  and  to  describe  it. 

Let  A,  B,  and  C  be  the  three  given  points 

1st.  Take  off  the  distance  B  A  between  the  intermediate 
point  and  one  of  the  exterior  points,  as  A,  with  the  dividers 
and  pencil  point,  and  with  this  opening  describe  two  arcs 
from  B,  on  either  side  of  B  A. 


44  INDUSTRIAL   DRAWING. 

2d.  With  the  same  opening  describe  from  A  two  like  arcs, 
and  mark  the  points  a  and  b  where  these  cross  the  two 
described  from  B. 

3d.  Draw  a  line  through  a  b. 

4th.  With  the  distance  B  C  in  the  dividers  describe,  from 
B  and  £7,  arcs  as  in  the  preceding  case,  and  mark  the  points 
c  and  d  where  these  cross  ;  and  then  draw  a  line  through  c  d. 

5th.  Having  marked  the  point  0  where  the  t  yo  lines  thus 
drawn  cross,  place  the  steel  point  of  the  dividers  at  0,  and 
extending  the  pencil  point  to  J.,  or  either  of  the  three  given 
points,  describe  an  arc,  or  a  complete  circle,  with  this  opening 
This  will  be  the  required  arc  or  circle. 

Verification.  The  fact  that  the  arc  or  circle  is  found  to 
pass  accurately  through  the  three  points  is  the  best  proof  of 
the  correctness  of  the  operations. 

Prob.  20.  (PL  II.  Fig.  26.)  At  a  point  on  an  arc  or  the 
circumference  of  a  circle  to  construct  a  tangent  to  the  arc  or  the 
nrde. 

Let  D  be  the  given  point  and  0  the  centre  of  the  circle. 

1st.  Draw  through  D  a  radius  0  D,  and  prolong  it  outwards 
from  the  arc. 

2d.  At  D  construct  by  Prob.  6  a  perpendicular  ED  Fto 
0  D.  This  is  the  required  tangent. 

If  the  centre  of  the  arc  or  the  circle  is  not  given,  proceed 
as  follows : — 

1st.  With  any  convenient  opening  in  the  dividers  and 
pencil  point  (Fig.  26)  set  off  from  D  the  same  arc  on  each 
side  of  it,  and  mark  the  points  A  and  B. 

2d.  Take  off  the  distance  A  B  and  describe  with  it  arcs 
from  A  and  B  on  each  side  of  the  given  arc,  and  mark  the 
points  a  and  b  where  they  cross. 

3d.  Draw  a  line  through  a  b. 

4th.  Construct  a  perpendicular  to  a  b  at  D.  This  is  the 
required  tangent. 

Verification.  Having  set  off  from  D  the  same  distance  on 
each  side  of  it  along  a  5,  and  having  set  off  also  any  distance 
from  D  along  E  F,  the  distance  from  this  last  point  to  the  other 
two  set  off  on  a  b  will  be  found  equal,  if  the  construction  is 
accurate. 


PROBLEMS  OF   CIRCLES,   &C.  45 

Remarks.  It  sometimes  happens  that  the  point  to  which  a 
tangent  is  required  is  so  near  the  extremity  of  the  arc,  as  at 
A,  or  (7,  that  the  method  last  explained  cannot  be  applied. 
In  such  a  case  we  must  first  find  the  centre  of  the  arc,  or 
circle,  which  will  be  done  by  marking  two  other  points,  as  B 
and  J.,  on  the  arc,  and  by  Prob.  19  finding  the  centre  0  of 
the  circle  of  which  this  arc  is  a  portion  of  the  circumference. 
Having  thus  found  the  centre,  the  tangent  at  0  will  be  con- 
structed by  the  first  method  in  this  Prob. 

Prob.  21.  (PL  II.  Fig.  27.)  At  a  given  point  on  the  cir- 
cumference of  a  given  circle,  to  construct  a  circle,  or  arc,  of  a 
given  radius  tangent  to  the  given  circle. 

Let  B  be  the  given  point,  and  D  the  centre,  which  is  either 
given,  or  has  been  found  by  Prob.  19. 

1st.  Through  DB  draw  the  radius,  which  extend  outwards 
if  the  centres  of  the  required  circle  and  of  the  given  one  are 
to  lie  on  opposite  sides  of  a  tangent  line  to  the  first  circle  at 
B ;  or,  in  the  contrary  case,  extend  it,  if  requisite,  from  D  in 
the  opposite  direction. 

2d.  From  B  set  off  along  this  line  the  length  of  the  given 
radius  of  the  required  circle  to  (7,  or  to  A. 

3d.  From  (7,  or  A,  with  the  distance  CB,  or  AB,  describe  a 
circle.  This  is  the  one  required. 

Prob.  22.  (PI.  II.  Fig.  28.)  From  a  given  point  without  a 
given  circle,  to  draw  two  tangents  to  the  circle. 

Let  A  be  the  centre  of  the  given  circle,  and  B  the  given 
point. 

1st.  Through  AB  draw  a  line. 

2d.  Divide  the  distance  AB  into  two  equal  parts  by 
Prob.  5. 

3d.  From  (7,  with  the  radius  CA,  describe  an  arc,  and  mark 
the  points  D,  and  E,  where  it  crosses  the  circumference  of  the 
given  circle. 

4th.  From  B  draw  lines  through  D  and  E.  These  lines 
are  the  required  tangents. 

Verification.  If  lines  are  drawn  from  A,  to  D  and  E}  they 
will  be  found  perpendicular  respectively  to  the  tangents,  if 
the  construction  is  accurate. 

Remark.    In  this  Prob.,  as  in  most  geometrical  construe- 


46  INDUSTRIAL   DRAWING. 

tions,  many  of  the  lines  of  construction  need  not  be  actually 
drawn,  either  in  whole,  or  in  part.  In  this  case,  for  example, 
a  small  portion  of  the  line  A — B,  at  its  middle  point,  is  alone 
necessary  to  determine  this  point.  In  like  manner,  the  points 
L  and  E  could  have  been  marked,  without  describing  the  arc 
actually,  but  by  simply  dotting  the  points  required.  In  this 
manner,  a  draftsman,  by  a  skilful  selection  of  his  lines  of 
construction,  and  using  only  such  of  them,  in  whole,  or  in 
part,  as  are  indispensably  requisite  for  the  solution,  may,  in 
complicated  constructions,  avoid  confusion  from  the  intersec- 
tion of  a  multiplicity  of  lines  of  construction,  and  abridge  his 
labor. 

Prob.  23.  (PI.  II.  Fig.  29.)  To  draw  a  tangent  to  two  given 
circles. 

Let  A  be  the  centre  of  one  of  the  circles,  and  A  Cits  radius; 
B  the  centre,  and  BE  the  radius  of  the  other. 

1st.  Through  AB  draw  a  line. 

2d.  From  C  set  off  CD  equal  to  BE. 

3d.  From  A,  with  the  radius  AD,  equal  to  the  difference 
between  the  radii  of  the  given  circles,  describe  a  circle. 

4th.  From  B  by  Prob.  22  draw  a  tangent  BD  to  this  last 
circle,  and  through  the  tangential  point  D,  a  radius  A  C  to  the 
given  circle. 

5th.  Through  C  draw  a  line  parallel  to  BD.  This  line 
will  touch  the  other  given  circle,  and  is  the  required  tangent. 

Verification.  CE  will  be  found  equal  to  BD,  if  the  con- 
struction is  accurate. 

Prob.  24.  (PI.  II.  Fig.  30.)  Having  two  lines  that  make  an 
angle,  to  construct  within  the  angle,  a  circle  with  a  given  radius 
tangent  to  the  two  given  lines. 

Let  AB  and  A  C  be  the  two  given  lines  containing  the 
angle. 

1st.  By  Prob.  16  construct  the  line  AD  which  divides  the 
given  angle  into  two  equal  parts. 

2d.  By  Prob.  10  set  off  a  point  b  at  a  distance  from  AB 
equal  to  the  given  radius,  and  through  this  point  draw  the 
line  ab  parallel  to  AB,  and  mark  the  point  a  where  it  crosses 
AD. 


PROBLEMS  OF   CIRCLES,   &C.  47 

8d.  From  a,  with  the  given  radius,  describe  a  circle.  This 
is  the  one  required. 

Verification.  The  distances  Ac,  and  Ad:  will  be  found 
equal,  if  the  construction  is  accurate. 

Prob.  25.  (PI.  II.  Fig.  31.)  Having  two  lines  containing 
an  angle,  and  a  given  radius  of  a  circle,  to  construct,  as  in  the 
last  case,  this  circle  tangent  to  the  two  lines  ;  and  then  to  construct 
another  circle  which  shall  be  tangent  to  the  last  and  also  to  the 
two  lines. 

Let  AB  and  CD  be  the  two  lines,  the  point  of  meeting  of 
which  is  not  on  the  drawing. 

1st.  By  Prob.  17  find  the  line  EFthzt  equally  divides  the 
angle  between  the  lines. 

2d.  By  Prob.  24-  construct  the  circle,  with  the  given  radius 
oj,  tangent  to  these  two  lines. 

3d.  At  b,  where  EF  crosses  the  circumference,  draw  by 
Prob.  20  a  tangent  to  the  circle,  and  mark  the  point  d  where 
it  crosses  AB. 

4th.  From  d,  set  off  de  equal  to  db,  and  mark  the 
point  e. 

5th.  At  e  construct  by  Prob.  6'  a  perpendicular  ef,  to  AB, 
and  mark  the  point/  where  it  crosses  EF. 

6th.  From/  with  the  radius  fe,  describe  a  circle.  This  is 
tangent  to  the  first  circle,  and  to  the  two  given  lines. 

Remark.  In  like  manner  a  third  circle  might  be  con- 
structed tangent  to  the  second  and  to  the  two  given  lines ; 
and  so  on  as  many  in  succession  as  may  be  wanted. 

Prob.  26.  (PI.  III.  Fig.  32.)  Having  a  circle  and  right  line 
given,  to  construct  a  circle  of  a  given  radius  which  shall  be  tangent 
to  the  given  circle  and  right  line. 

Let  G  be  the  centre  of  the  given  circle,  and  AB  the  given 
line. 

1st.  By  Prob.  12  draw  a  line  parallel  to  AB,  and  at  a 
distance  BG  from  it,  equal  to  the  given  radius. 

2d.  Draw  a  radius  CD  through  any  point  D  of  the  given 
circle,  and  prolong  it  outwards. 

3d.  From  D  set  off,  along  the  radius,  a  distance  DE  equal 
to  BG,  or  the  given  radius. 

4th.    From  C,   with  the    distance    CE,    describe  an   arc 


48  INDUSTRIAL  DRAWING. 

and  mark  the  point  F  where  it  crosses  the  line  parallel  to 
AB. 

5th.  From  the  point  F,  with  the  given  radius  describe  a 
circle.  This  is  the  one  required. 

Remarks.  If  the  construction  is  accurate  a  line  drawn  from 
F  to  G  will  pass  through  the  point  where  the  circles  touch, 
and  one  drawn  from  ^perpendicular  to  AB  will  pass  through 
the  point  where  the  circle  touches  the  line. 

If  from  the  centre  (?,  a  perpendicular  is  drawn  to  AB,  and 
the  points  a,  I  and  d  where  the  perpendicular  crosses  the  line 
and  the  given  circle  are  marked,  it  will  be  found  that  the 
given  radius  cannot  be  less  than  one  half  of  ab  nor  greater 
than  one  half  of  ad. 

Prob.  27.  (PI.  III.  Fig.  33.)  Having  a  circle  and  right  line 
given,  to  construct  a  circle  which  shall  he  tangent  to  the  given 
circle  at  a  given  point,  and  also  to  the  line. 

Let  C  be  the  centre  of  the  given  circle,  D  the  given  point 
on  its  circumference,  and  AB  the  given  line. 

1st.  By  Prob.  20  construct  a  tangent  to  the  given  circle  at 
the  point  D;  prolong  it  to  cross  the  given  line,  and  mark  the 
point  A  where  it  crosses. 

2d.  By  Prob.  16  construct  the  line  AE  which  bisects  the 
angle  between  the  tangent  and  the  given  line. 

3d.  Through  CD  draw  a  radius,  and  prolong  it  to  cross  the 
bisecting  line  at  E. 

4th.  Mark  the  point  E,  and  with  the  distance  ED  describe 
a  circle.  This  is  the  required  circle. 

Prob.  28.  (PI.  III.  Fig.  34.)  Having  a  circle  and  right  line, 
to  construct  a  circle  which  shall  he  tangent  to  the  given  circle,  and 
also  to  the  line  at  a  given  point  on  it. 

Let  C  be  the  centre  of  the  given  circle ;  AB  the  given  line, 
and  a  the  given  point  on  it. 

1st.  By  Prob.  6  construct  a  perpendicular  at  a  to  the  given 
line. 

2d.  From  a  set  off  ah  equal  to  the  radius  Cd  of  the  given 
circle. 

3d.  Draw  a  line  through  bC,  and  by  Prob.  5  bisect  the 
distance  bC  by  a  perpendicular  to  the  line  bC. 

4th.  Mark  the  point  c,  where  this  perpendicular  crosses  the 


PROBLEMS   OF   CIRCLES,    &C.  49 

one  at  a;  and  with  ac  describe  a  circle.     This  is  the  one 
required. 

Prob.  29.  (PI.  III.  Fig.  35.)  Having  two  circles,  to  construct 
a  third  which  shall  be  tangent  to  one  of  them  at  a  given  point, 
and  also  touch  the  other. 

Let  C  and  B  be  the  centres  of  the  two  given  circles ;  and 
D  the  given  point  on  one  of  them,  at  which  the  required 
circle  is  to  be  tangent  to  it. 

1st.  Through  CD  draw  a  line,  which  prolong  each  way 
from  C  and  D. 

2d.  From  D  set  off  towards  0  the  distance  D  A  equal  to  the 
radius  BF  of  the  other  given  circle. 

3d.  Through  AB  draw  a  line,  and  by  Probs.  5  and  6,  bisect 
AB  by  the  perpendicular  GK 

4th.  Mark  the  point  E,  where  the  perpendicular  crosses 
the  line  CD  prolonged ;  and  with  the  distance  ED  describe  a 
circle  from  K  This  is  the  one  required. 

Prob.  30.  (PI.  III.  Fig.  36.)  Having  a  given  distance,  or 
line,  and  the  perpendicular  which  bisects  it,  to  construct  three 
arcs  of  circles,  the  radii  of  two  of  which  shall  be  equal,  and  of  a 
given  length,  and  their  centres  on  the  given  line  ;  and  the  third 
shall  pass  through  a  given  point  on  the  perpendicular  and  be 
tangent  to  the  other  two  circles. 

Let  AB  be  the  given  line,  and  D  the  given  point  on  the 
perpendicular  to  AB  through  its  middle  point  C  ;  and  let  the 
distance  CD  be  less  than  A  C,  the  half  of  AB. 

1st.  Take  any  distance,  less  than  CD,  and  set  it  off  from  A 
and  B,  to  b  and  e,  and  mark  these  two  points  for  the  centres 
of  the  two  arcs  of  the  equal  given  radii  less  than  CD. 

2d.  Set  off  from  D  the  distance  DC  equal  to  Ab,  and  through 
be  draw  a  line. 

3d.  Bisect  be  by  a  perpendicular,  by  Probs.  5  and  6,  and 
mark  the  point  d  where  it  crosses  the  perpendicular  to  AB 
prolonged  below  it,  the  point  d  is  the  centre  of  the  third  arc. 

4th.  Draw  a  line  through  db,  and  prolong  it ;  and  also  one 
through  de  which  prolong. 

5th.  From  b,  with  the  distance  bA,  describe  an  arc  from  A 
to  m  on  the  line  db  prolonged ;    and  one,  from  the  othei 
centre  e,  from  B  to  n  on  the  line  de  prolonged. 
4 


60  INDUSTRIAL   DRAWING. 

6th.  From  d,  with  the  distance  dD,  describe  an  arc  around 
to  the  two  lines  db,  and  de  prolonged.  This  is  the  third  arc 
required,  and  touches  the  other  two  where  they  cross  the 
fines  db  and  de  prolonged  at  m  and  n. 

Remark.  This  curve  is  termed  a  half  oval,  or  a  three  centre 
curve.  The  other  half  of  the  curve,  on  the  other  side  of  AB, 
can  be  drawn  bj  setting  off  a  distance  Cg  equal  to  Cd,  and 
by  continuing  the  arcs  described  from  b  and  e  around  to  lines 
drawn  from  g,  through  b  and  e ;  and  by  connecting  these  arcs 
by  another  described  from  g,  with  a  radius  equal  to  Dd. 

Prob.  81.  (PI.  III.  Fig.  37.)  Having  a  given  line  and  the 
perpendicular  that  bisects  it;  also  two  lines  drawn  through  a 
given  point,  on  the  perpendicular,  and  each  making  the  same 
angle,  with  it;  to  construct  a  curve  formed  of  four  arcs  of  circles, 
two  of  these  arcs  to  have  equal  given  radii,  and  their  centres  to  lie 
on,  the  given  line,  and  at  equal  distances  from  its  extremities  ;  each 
of  the  other  arcs  to  have  equal^  radii,  and  to  be  tangent  respec- 
tively to  one  of  the  given  lines  where  it  crosses  the  perpendicular 
and  also  to  one  of  the  first  arcs. 

Let  CD  be  the  given  line ;  B  the  given  point  on  the  bisect- 
ing perpendicular;  and  Bm,  Bn,  the  two  lines,  drawn 
through  B,  making  the  same  angle  with  the  perpendicular. 

1st.  From  0  and  D,  set  off  the  same  distance  to  b  and  a,  for 
the  given  radii  of  the  two  first  arcs ;  which  distance  must, 
in  all  cases,  be  taken  less  than  the  perpendicular  distance  from 
the  point  b  or  a,  to  one  of  the  given  lines  through  B. 

2d.  At  B  draw  a  perpendicular  to  the  line  Bm. 

3d.  Set  off  from  B,  along  this  perpendicular,  a  distance  Bd 
equal  Cb. 

4th.  Draw  a  line  through  bd,  and  bisect  this  distance  by  a 
perpendicular. 

5th.  Having  marked  the  point  c,  where  this  last  perpen- 
dicular crosses  the  one  at  B,  draw  through  cb  a  line,  and 
prolong  it  beyond  b. 

6th.  From  b,  with  the  distance  1C,  describe  an  arc,  which 
prolong  to  the  line  through  be  ;  this  is  one  of  the  first  required 
arcs. 

7th.  From  c,  with  a  distance  cB,  describe  an  arc.  This  if 
one  of  the  second  required  arcs. 


PROBLEMS   OF   CIRCLES,    &C.  51 

8th.  Through  a,  and  the  point/  where  be  crosses  the  per 
pendicular  BA  prolonged,  draw  a  line. 

9th.  From  a,  set  off  ae  equal  to  be. 

10th.  From  a  and  e,  with  radii  respectively  equal  to  bC,  and 
cB,  describe  arcs.  These  are  the  others  required ;  and  CBD 
the  required  curve. 

Remark.  If  the  construction  is  accurate,  the  perpendicular 
through  BA  will  bisect  the  distance  ec. 

This  curve  is  termed  a  four  centre  obtuse  or  pointed  curve, 
according  as  the  distance  AB  is  less  or  greater  than  A  G. 

Prob.  32.  (PL  III.  Fig.  38.)  Having  a  line,  and  the  per- 
pendicular which  bisects  it,  and  a  given  point  on  the  perpen- 
dicular ;  to  construct  a  curve  formed  of  jive  arcs  of  circles,  the 
consecutive  arcs  to  be  tangent ;  the  centres  of  two  of  the  arcs  to  be 
on  the  given  line,  and  at  equal  distances  from  its  extremities;  the 
radii  of  the  two  arcs,  respectively  tangent  to  these  two,  to  be  equal, 
and  of  a  given  length ;  and  the  centre  of  the  fifth  arc,  which  is 
to  be  tangent  to  these  two  last,  to  lie  on  the  given  perpen 
dicular. 

Let  AB  be  the  given  line,  and  C  the  point  on  its  bisecting 
perpendicular  LC. 

1st.  Take  any  distance,  less  than  LCy  and  set  it  off  from  B 
to  D,  and  from  A  tof. 

2d.  From  C  set  off  CG  equal  to  BD,  and  draw  a  line  from 
GtoD. 

3d.  Bisect  the  distance  DG  by  a  perpendicular ;  and  mark 
the  point  E,  where  this  perpendicular  crosses  the  perpen- 
dicular L  C  prolonged. 

4th.  Draw  a  line  from  E  to  D. 

5th.  Take  any  distance,  less  than  CE,  equal  to  the  given 
radius  of  the  second  arc,  and  set  it  off  from  G  to  F. 

6th.  through  F  draw  a  line  FH  parallel  to  AB. 

7th.  Take  off  GF,  the  difference  between  CF  and  GQ,  and 
with  it  describe  an  arc  from  D  ;  and  mark  the  points  a  and  b 
where  it  crosses  the  lines  DE  and  FH. 

8th.  Take  any  point  c  on  this  arc,  between  the  points  a 
and  b,  and  draw  from  it  a  line  to  F. 

9th.  Bisect  the  line  cF  by  a  perpendicular,  and  mark  the 
point  /where  it  crosses  the  perpendicular  GL  prolonged. 


52  INDUSTRIAL  DRAWING. 

10th.  From  c  draw  a  line  through  D  and  prolong  it ;  and 
another  from  /prolonged  through  c. 

llth.  From  c  draw  a  perpendicular  to  CI :  and  from  d, 
where  it  crosses  CI,  set  off  dg  equal  to  cd,  and  mark  the 
point  g. 

12th.  From  /draw  a  line  through  g  and  prolong  it;  and 
one  from  g  prolonged  through  / 

13.  From  D  and/  with  the  distance  BD,  describe  the  arcs 
Bm,  and  Ap ;  from  c  and  g,  with  the  distance  cm,  or  gp, 
describe  the  arcs  mn,  and  po ;  and  from  /  with  the  distance 
1C,  describe  the  arc  no.  The  curve  BCA  is  the  one 
required. 

Remarks.  This  curve  is  also  termed  a  semi  oval;  and, 
from  the  number  of  arcs  of  which  it  is  composed,  a  curve  of 
Jive  centres. 

Prob.  33.  (PI.  III.  Fig.  39.)  Having  two  parallel  lines,  to 
construct  a  curve  of  three  centres  which  shall  be  tangent  to  the  two 
parallels  at  their  extremities. 

Let  AB  and  CD  be  the  given  parallels  ;  and  B  and  D  the 
points  at  which  the  required  curve  is  to  be  drawn  tangent. 

1st.  From  B  construct  a  perpendicular  to  AB,  and  mark 
the  point  b  where  it  crosses  CD;  and  also  a  perpendicular  at 
D  to  CD. 

2d.  From  B,  set  off  any  distance  Be  less  than  the  half  of 
Bb  •  and  through  c  draw  a  line  parallel  to  AB,  and  mark  the 
point  d  where  it  crosses  the  perpendicular  to  CD. 

3d.  From  c,  set  off,  along  cd  prolonged,  the  distance  ca 
equal  to  cB. 

4th.  Taking  ca,  as  the  radius  of  the  first  arc,  construct  a 
quarter  oval  by  Prob.  30  through  the  points  a  and  D. 

5th.  Prolong  the  arc  described  from  c  to  the  point  B.  The 
curve  BaD  is  the  one  required. 

Remark.     This  curve  is  termed  a  scotia  of  two  centres. 

Prob.  34.  (PI.  III.  Fig.  40.)  Having  two  parallels,  to  con- 
struct a  quarter  of  a  curve  of  five  centres  tangent  to  them  at  their 
extremities. 

Let  AB  and  CD  be  the  given  parallels,  and  B  and  D  theiz 
extremities. 

1st.  Proceed,  as  in  the  last  case,  to  draw  the  perpendiculars 


PROBLEMS  OF  CIRCLES,   &C.  53 

at  B,  and  D,  and  a  parallel  to  AS  through  a  point  c,  taken 
on  the  first  perpendicular,  at  a  distance  from  B  less  than  the 
half  of  m 

2d.  Set  off  from  c  a  distance  ca  equal  to  cB ;  and  on  cut 
and  Dd  describe  the  quarter  oval  by  Prob.  32. 

3d.  Prolong  the  first  arc  from  a  to  B,  which  will  complete 
the  required  curve. 

Remark.     This  curve  is  termed  a  scotia  of  three  centres. 

Prob.  35.  (PI.  III.  Fig.  41.)  Having  two  parallels  and  a 
given  point  on  each,  to  construct  two  equal  arcs  which  shall  be 
tangent  to  each  other  and  respectively  tangent  to  the  parallels  ai 
the  given  points. 

Let  AB  and  CD  be  the  two  parallels ;  B  and  D  the  given 
points. 

1st.  Draw  a  line  through  BD,  and  bisect  the  distance 
BD. 

2d.  Bisect  each  half  BE,  and  ED  by  perpendiculars. 

3d.  From  B,  and  D  draw  perpendiculars  to  AB  and  CD, 
and  mark  the  points  a,  and  b,  where  they  cross  the  bisecting 
perpendiculars.  - 

4th.  From  a,  with  the  distance  aE,  describe  an  arc  to  B; 
and  from  b,  with  the  same  distance,  an  arc  ED,  These  are 
the  required  arcs. 

Prob.  36.  (PI.  III.  Fig.  42.)  Having  two  parallels,  and  a 
point  on  each,  to  construct  two  equal  arcs  which  shall  be  tangent 
to  each  other,  have  their  centres  respectively  on  the  parallels, 
and  pass  through  the  given  points. 

Let  AB  and  CD  be  the  parallels,  B  and  D  the  given 
points. 

1st.  Join  BD  by  a  line  and  bisect  it. 

2d.  Bisect  each  half  BE,  and  ED  by  a  perpendicular ;  and 
mark  the  points  a  and  b,  where  the  perpendiculars  cross  the 
parallels. 

3d.  From  a,  with  aB,  describe  the  arc  BE;  and  from  b, 
with  the  same  distance,  the  arc  DE.  These  are  the  required 
arcs. 


54  INDUSTRIAL  DRAWING. 


CONSTRUCTION  OF  PROBLEMS  OF  CIRCLES  AND 
RECTILINEAL  FIGURES. 

Piob.  37.  (PL  III.  Fig.  43.)  Having  the  sides  of  a  trianglt 
to  construct  the  figure. 

Let  A  C,  BO,  and  AB  be  the  lengths  of  the  given  sides. 

1st.  Draw  a  line,  and  set  off  upon  it  the  longest  side  AB. 

2d.  From  the  point  J.,  with  a  radius  equal  to  A  C,  one  ol 
the  remaining  sides,  describe  an  arc. 

3d.  From  the  point  B,  with  the  third  side  BC,  describe  a 
second  arc,  and  mark  the  point  G  where  the  arcs  cross. 

4th.  Draw  lines  from  C,  to  A  and  B.  The  figure  A  CB  is 
the  one  required. 

Remark.  The  side  AC  might  have  been  set  off  from  B, 
and  BC  from  A ;  this  would  have  given  an  equal  triangle  to 
the  one  constructed,  but  its  vertex  would  have  been  placed 
differently. 

Remark.  This  construction  is  also  used  to  find  the  position 
of  a  point  when  its  distances  from  two  other  given  points  are 
given.  We  proceed  to  make  the  construction  in  this  case 
like  the  preceding.  It  will  be  seen,  that  the  required  point 
can  take  four  different  positions  with  respect  to  the  two 
others.  Two  of  them,  like  the  vertex  of  the  triangle,  will  lie 
on  one  side  of  the  line  joining  the  given  points,  and  the  other 
two  on  the  other  side  of  the  line. 

Prob.  38.  (PI.  III.  Fig.  44.)  Having  the  side  of  a  square  to 
construct  the  figure. 

Let  AB  be  the  given  side. 

1st.  Draw  a  line,  and  set  off  AB  upon  it. 

2d.  Construct  perpendiculars  at  A,  and  J5,  to  AB. 

3d.  From  A  and  B,  set  off  the  given  side  on  these  perpen- 
diculars to  G  and  D  ;  and  draw  a  line  from  G  to  D.  The 
figure  ABCD  is  the  one  required. 

Prob.  39.  (PL  III.  Fig.  45.)  Having  the  two  sides  of  a 
parallelogram,  and  the  angle  contained  by  them,  to  construct  the 
figure. 

Let  AB,  and  AC  be  the  given  sides;  and  E  the  given 
angle. 


PROBLEMS  OF   CIRCLES,   &C.  55 

1st.  Draw  a  line,  and  set  off  AB  upon  it. 

2d  Construct  at  A  an  angle  equal  to  the  given  one  bj 
Prob.  13. 

3d.  Set  offj  along  the  side  of  this  angle,  from  A,  the  othei 
given  line  A  0. 

4th.  From  C,  with  the  distance  AB  describe  an  arc,  and 
from  B  with  the  distance  A  C  describe  another  arc. 

5th.  From  the  point  D,  where  the  arcs  cross,  draw  lines  to 
(7,  and  B.  The  figure  ABDCis  the  one  required. 

Prob.  40.  (PI.  III.  Fig.  46.)  To  circumscribe  a  given  triangle 
by  a  circle. 

Let  ABO  be  the  given  triangle. 

As  the  circumference  of  the  required  circle  must  be 
described  through  the  three  given  points  A,  B  and  (7,  its 
centre  and  radius  will  be  found  precisely  as  in  Prob.  19. 

Prob.  41.  (PL  III.  Fig.  47.)  In  a  given  triangle  to  inscribe 
a  circle. 

Let  ABC  be  the  given  triangle. 

1st.  By  Prob.  16  construct  the  lines  bisecting  the  angles  A, 
and  G ;  and  mark  the  point  D  where  these  lines  cross. 

2d.  From  D  by  Prob.  8  construct  a  perpendicular  DE,  to 
AC. 

3d.  From  D,  with  the  distance  DE,  describe  a  circle.  This 
is  the  one  required. 

Prob.  42.  (PI.  IY.  Fig.  48.)  In  a  given  circle  to  inscribe  a 
square. 

Let  0  be  the  centre  of  the  given  circle. 

1st.  Through  0  draw  a  diameter  AB,  and  a  second  dia- 
meter CD  perpendicular  to  it. 

2d.  Draw  the  lines  AC,  CB,  ED,  and  DA.  The  figure 
ADBC  is  the  one  required. 

Prob.  43.  (PI.  IV.  Fig.  48.)  In  a  given  circle  to  inscribe  a 
regular  octagon. 

1st.  Having  inscribed  a  square  in  the  circle  bisect  each  of 
its  sides ;  and  through  the  bisecting  points  and  the  centre  G 
draw  radii. 

2d.  Draw  lines  from  the  points  cZ,  5,  a,  c,  where  these  radii 
meet  the  circumference,  to  the  adjacent  points  D^  A,  &c.  The 
figure  dAbC&c.  is  the  one  required. 


56  INDUSTRIAL  DRAWING. 

Remark.  By  bisecting  the  sides  of  the  octagon,  and  draw 
ing  radii  through  the  points  of  bisection,  and  then  drawing 
lines  from  the  points  where  these  radii  meet  the  circumference 
to  the  adjacent  points  of  the  octagon,  a  figure  of  sixteen  equa] 
sides  can  be  inscribed,  and  in  like  manner  one  of  32  sides, 
&c. 

Prob.  44.  (PL  IY.  Fig.  49.)  To  inscribe  in  a  given  circk 
a  regular  hexagon. 

Let  0  be  the  centre  of  the  given  circle. 

1st.  Having  taken  off  the  radius  OA,  commence  at  A,  and 
Bet  it  off  from  A  to  Bt  and  from  A  to  F^  on  the  circum- 
ference. 

2d.  From  B  set  off  the  same  distance  to  C ;  and  from  C  to 
D,  and  so  on  to  F. 

3d.  Draw  lines  between  the  adjacent  points.  The  figure 
AEG  &c.  is  the  one  required. 

Remark.  By  a  process  similar  to  the  one  employed  for 
constructing  an  octagon  from  a  square,  we  can,  from  the 
hexagon,  construct  a  figure  of  12  sides ;  then  one  of  24 ;  and 
so  on  doubling  the  number  of  sides. 

Prob.  45.  (PI.  IV.  Fig.  49.)  To  inscribe  in  a  given  circle  an 
equilateral  triangle. 

Having,  as  in  the  last  problem,  constructed  a  regular  hex- 
agon, draw  lines  between  the  alternate  angles,  as  A  (7,  CE, 
and  EA  ;  the  figure  thus  formed  is  the  one  required. 

Prob.  46.  (PL  IY.  Fig.  50.)  To  inscribe  in  a  given  circle  a 
regular  pentagon. 

Let  0  be  the  centre  of  the  given  circle. 

1st.  Draw  a  diameter  of  the  circle  AB,  and  a  second  one 
CD  perpendicular  to  it. 

2d.  Bisect  the  radius  OB,  and  from  the  point  of  bisection 
2  set  off  the  distance  a  Ct  to  5,  along  AB. 

3d.  From  C,  with  the  radius  Cb,  describe  an  arc,  and  mark 
the  points  H,  and  It  where  it  crosses  the  circumference  of  the 
given  circle. 

4th.  From  H,  and  /,  set  off  the  same  distance  to  G  and  J£ 
on  the  circumference. 

5th.  Draw  the  lines  Off,  HG,  GK,  KI,  and  1C.  The 
figure  CHGKIis  the  one  required. 


PROBLEMS  OF   CIRCLES,    &C.  57 

Prob.  47.     To  construct  a  regular  figure,  the  sides  of  which 
shall  be  respectively  equal  to  a  given  line. 
Let  AB  be  the  given  line. 

First  Method.    (PI.  IV.  Fig.  50.) 

1st.  Construct  any  circle,  and  inscribe  within  it  a  regular 
figure,  by  one  of  the  preceding  Probs.  of  the  same  number  of 
sides  as  the  one  required. 

Let  us  suppose  for  example  that  the  one  required  is  a  pen- 
tagon. 

2d.  Having  constructed  this  inscribed  figure,  draw  from 
the  centre  of  the  circle,  through  the  angular  points  of  the 
figure,  lines  ;  and  prolong  them  outwards,  if  the  side  of  the 
inscribed  figure  is  less  than  the  given  line. 

3d.  Prolong  any  one  of  the  sides,  as  Of,  of  the  inscribed 
figure,  and  set  off  along  it,  from  the  angular  point  C,  a  dis- 
tance Cm  equal  to  the  given  line. 

4th.  Through  m,  draw  a  line  parallel  to  the  line  drawn 
from  0  through  0,  and  mark  the  point  n,  where  it  crosses 
the  line  drawn  from  0  through  /. 

5th.  Through  n,  draw  a  line  parallel  to  CI,  and  mark  the 
point  o,  where  it  crosses  the  line  OC  prolonged. 

6th.  From  0,  set  off,  along  the  other  lines  drawn  from  G 
through  the  other  angular  points,  the  distances  Op,  Oq,  and 
Or,  each  equal  to  Om,  or  On. 

7th.  The  points  o,  p,  r,  and  n  being  joined  by  lines ;  the 
figure  opqrn  is  the  one  required. 

Second  Method.     (PI.  IV.  Fig.  51.) 

1st.  Draw  a  line  and  set  off  the  given  line  AB  upon  it. 

2d.  At  B  construct  a  perpendicular  to  AB. 

3d.  From  B,  with  BA,  describe  an  arc  Aa. 

4th.  Divide  this  arc  into  as  many  equal  parts  as  number  of 
sides  in  the  required  figure ;  and  mark  the  points  of  division 
from  a,  1,  2,  3,  &c. 

5th.  From  A,  with  AB,  describe  an  arc,  and  mark  the 
point  c  where  it  crosses  the  arc  Aa. 

6th   From  B  draw  a  line  through  the  division  point  2. 

7th.  From  c,  set  off  the  distance  c2,  to  b,  on  the  arc  Be. 


58  INDUSTRIAL   DRAWING. 

8th.  From  A,  draw  a  line  through  b,  and  mark  the  point 
0  where  it  crosses  £2. 

9th.  From  0,  with  the  distance  OA,  or  OB,  describe  a 
circle. 

10th.  Set  off  the  distance  AB  to  C,  D,  &c.,  on  the  circum- 
ference. 

llth.  Draw  the  lines  £0,  CD,  DE,  &c.  This  is  the 
required  figure. 

Remarks.  The  figure  taken  to  illustrate  this  case  is  the 
pentagon,  for  the  purpose  of  comparing  the  two  methods. 

Prob.  48.  (PI.  IV.  Fig.  51.)  To  circumscribe  a  given  circle 
by  a  regular  figure. 

1st.  Inscribe  in  the  circle  a  regular  figure  of  the  same 
number  of  sides  as  the  one  to  be  circumscribed. 

2d.  At  the  angular  points  of  the  inscribed  figure,  draw 
tangents  to  the  given  circle,  and  mark  the  points  where  the 
tangents  cross.  These  points  are  the  angular  points  of  the 
required  figure,  and  the  portions  of  the  tangents  between 
them  are  its  sides. 

Remarks.  The  figure'  taken  to  illustrate  this,  is  the  cir- 
cumscribed regular  pentagon  bcdef. 

Prob.  49.  (PL  IY.  Fig.  49.)  To  inscribe  a  circle  in  a  given 
regular  figure. 

1st.  Bisect  any  two  adjacent  sides  of  the  figure  by  perpen- 
diculars, and  mark  the  point  where  they  cross. 

2d.  From  this  point,  with  the  distance  to  the  side  bisected, 
describe  a  circle.  This  is  the  one  required. 

Remarks.  The  figure  taken  to  illustrate  this  case  is  the 
regular  hexagon  ;  mn  and  np  are  the  adjacent  sides  bisected 
by  the  perpendiculars  to  them  aO,  and  bO  ;  0  is  the  centre 
of  the  required  circle,  and  Oa  its  radius. 

Prob.  50.  (PL  IV.  Fig.  52.)  To  inscribe,  in  a  given  circle, 
a  given  number  of  equal  circles  which  shall  be  tangent  to  the  given 
circle,  and  to  each  other. 

Let  0  be  the  centre  of  the  given  circle. 

1st.  Divide  the  circumference  into  as  many  equal  parts,  by 
lines  drawn  from  0,  as  the  number  of  circles  to  be 
inscribed.  Let  us  take,  for  illustration,  six  as  the  required 
number. 


PROBLEMS  OF   CIRCLES,   &C.  59 

2d.  Bisect  the  angle,  as  DOB,  between  any  two  of  these 
lines  of  division,  and  prolong  out  the  bisecting  line. 

3d.  Construct  a  tangent  to  the  given  circle  at  either  B,  or 
7),  and  mark  the  point  a  where  this  tangent  crosses  the 
bisecting  line. 

4th.  From  a,  set  off  aB  to  J,  along  the  bisecting  line. 

5th.  A't  b  construct  a  perpendicular  to  Oa,  and  mark  the 
point  c  where  it  crosses  OB. 

6th.  From  c,  with  the  distance  cBt  describe  a  circle.  Thia 
is  one  of  the  required  circles. 

7th.  From  the  other  points  of  division,  D,  &c.,  set  off 
the  same  distance  Be,  and  from  the  points  thus  set  off  with 
this  distance  describe  circles.  These  are  the  other  required 
circles. 

Prob.  51.  (PI.  IV.  Fig.  52.)  To  circumscribe  a  given  circle 
by  a  given  number  of  circles  tangent  to  it,  and  to  each  oilier. 

1st.  Having  divided  the  given  circle  into  a  number  of 
equal  parts,  the  same  as  the  given  number  of  required  circles: 
bisect,  in  the  same  way,  the  angle  between  any  two  adjacent 
lines  of  division. 

Let  us  take  for  illustration  six  as  the  number  of  required 
circles. 

2d.  Prolong  outwards  one  of  the  lines  of  division,  as  OD, 
and  also  the  line,  as  Od,  that  bisects  the  angle  between  it  and 
the  adjacent  line  of  division  Oa.  Construct  a  tangent  aiD 
to  the  given  circle  ;  and  mark  the  point /where  it  crosses  the 
bisecting  line. 

3d.  From/  set  off fD' to  d,  along  the  bisecting  line ;  and 
at  d,  construct  a  perpendicular  to  this  line,  and  mark  the  point 
g  where  it  crosses  the  line  OD'- 

4th.  From  g,  with  the  distance  gD\  or  gd,  describe  a  circle. 
This  is  one  of  the  required  circles. 

5th.  Prolong  outwards  the  other  lines  of  division ;  and  set 
off  along  them,  from  the  points  where  they  cross  the  circum- 
ference, the  distance  D'g ;  and  from  these  points  with  this 
distance  describe  circles.  These  are  the  remaining  required 
circles. 


60  INDUSTRIAL  DRAWING. 


CONSTRUCTION  OF  PROPORTIONAL  LINES  AND  FIGURES. 

Prob.  52.  (PL  IV.  Fig.  53.)  To  divide  a  given  line  into 
parts  which  shall  be  proportional  to  two  other  given  lines. 

Let  AB  be  the  given  line  to  be  divided ;  ac  and  cb  the 
other  given  lines. 

1st.  Through  A  draw  any  line  making  an  angle  with 
AB. 

2d.  From  A  set  off  Ac  equal  to  ac  :  and  from  c  the  other 
line  cb. 

3d.  Draw  a  line  through  B,  b  ;  and  through  c  a  parallel  to 
Bb,  and  mark  the  point  0  where  it  crosses  AB.  This  is  the 
required  point  of  division ;  and  A  C  is  to  CB  as  ac  is  to 
cb. 

Prob.  53.  (PI.  IV.  Fig.  54.)  To  divide  a  line  into  any 
number  of  parts  which  shall  be  in  any  given  proportion  to  each 
other,  or  to  the  same  number  of  given  lines. 

Let  AB  be  the  given  line,  and  let  the  number  of  pro- 
portional parts  for  example  into  which  it  is  to  be  divided 
be  four,  these  parts  being  to  each  other  as  the  numbers  3, 5,  7, 
and  2,  or  lines  of  these  lengths. 

1st.  Through  A  draw  any  line  making  an  angle  with 
AB. 

2d.  From  any  scale  of  equal  parts  take  off  three  divisions, 
and  set  this  distance  off  from  A  to  3  ;  from  3  set  off  five  of 
the  same  divisions  to  5  ;  from  5  set  off  seven  to  7;  and  from 
7  two  to  2. 

3d.  Draw  a  line  through  B2,  and  parallels  to  it  through 
the  points  7,  5,  and  3,  and  mark  the  points  d,  c,  and  b  where 
the  parallels  cross  AB.  The  distances  Ab,  be,  cd,  and  dB  are 
those  required. 

Remark.  Any  distance  from  a  point,  as  A  for  example,  on 
A  B,  to  any  other  point  as  d,  is  to  the  distance  from  this  point 
to  any  other,  as  Ab  for  example,  as  is  the  corresponding 
distance  A7  to  AB,  on  the  line  A2. 

Prob.  54.  (PI.  IV.  Fig.  55.)  To  find  a  fourth  proportional 
to  three  given  lines. 

Let  ab,  be,  and  ad  be  the  three  given  lines  to  which  it  is 


PROBLEMS   OF   PROPORTIONAL   LINES,    &C.  61 

required  to  find  a  fourth  proportional  which  shall  be  to  ad  as 
ab  is  to  ac. 

1st.  Draw  a  line,  and,  from  a  point  A,  set  off  AB  equal  tc 
ab  ;  and  BC  equal  to  be. 

2d.  Through  A  draw  any  line,  and  set  off  upon  it  AD, 
equal  to  ad. 

3d.  Draw  a  line  through  DB,  and  a  parallel  to  DB  through 
C,  and  mark  the  point  E  where  this  crosses  the  line  drawn 
through  A.  The  distance  DE  is  the  required  fourth  pro- 
portional. 

Prob.  55.     (PL  IV.  Fig.  56.)     To  find  the  line  which  is  a 
mean  proportional  to  two  given  lines. 
Let  ab  and  be  be  the  given  lines. 

1st.  Draw  a  line,  and  set  off  on  it  AB}  and  BC,  equal 
respectively  to  ab,  and  be. 

2d.  Bisect  the  distance  A  Cj  and,  from  the  bisecting  point 
0,  describe  a  semicircle  with  the  radius  OC. 

3d.  At  B  construct  a  perpendicular  to  AC;  and  mark  the 
point  D  where  it  crosses  the  circumference.  The  distance 
BD  is  the  line  required ;  and  ab  is  to  BD  as  BD  is  to 
be. 

Prob.  56.     (PI.  IV.  Fig.  57.)     To  divide  a  given  line  into 
two  parts,  such  that  the  entire  line  shall  be  to  one  of  the  parts,  as 
this  part  is  to  the  other. 
Let  ab  be  the  given  line. 

1st.  Draw  a  line,  and  set  off  AB  equal  to  ab;  and  at  B 
construct  a  perpendicular  to  AB. 

2d.  Set  off  on  the  perpendicular  BD  equal  to  the  half  of 
AB,  and  draw  a  line  through  AD. 

3d.  From  D,  with  DB,  describe  an  arc,  and  mark  the  point 
(7,  where  it  crosses  AD. 

4th.  From  A,  with  AC,  describe  an  arc,  and  mark  the  point 
E,  where  it  crosses  AB.  The  point  E  is  the  one  required ; 
and  AB  is  to  AE,  as  AE  is  to  EB. 

Remark.  This  construction  is  used  for  inscribing  a  regulai 
decagon  in  a  given  circle.  To  do  this  divide  the  radius  of  the 
given  circle  in  the  manner  just  described.  The  larger  portion 
is  the  side  of  the  required  regular  decagon. 

Having  described  the  regular  decagon,  the  regular  pen 


62  INDUSTEIAL   DRAWING. 

tagoii  can  be  formed,  by  drawing  lines  through  the  alternate 
angles  of  the  decagon. 

Prob.  57,  (PL  IV.  Fig.  58.)  Having  any  given  figure,  to 
construct  another,  the  angles  of  which  shall  be  the  same  as  tht, 
angles  of  the  given  figure,  and  the  sides  shall  be  in  a  given  pro- 
portion to  its  sides. 

Let  ABGDEF  be  the  given  figure. 

1st.  Prolong  any  two  of  the  adjacent  sides  of  the  given 
figure,  as  AB,  and  AF,  if  the  one  required  is  to  be  greater 
than  the  given  one ;  and,  from  A,  draw  lines  through  the 
other  angular  points  C,  D,  and  E. 

2d.  From  A  set  off  a  distance  Ab,  which  is  in  the  same 
proportion  to  AB,  as  the  side  of  the  required  figure  corres- 
ponding to  BC,  is  to  J3C ;  or,  in  other  words,  AB  must  be 
contained  as  many  times  in  Ab  as  JBG  is  in  the  corresponding 
side  of  the  required  figure. 

3d.  From  b  draw  a  line  parallel  to  BC,  and  mark  the  point 
c,  where  it  crosses  A  G  prolonged ;  from  c  draw  a  parallel  to 
CD,  and  mark  the  point  where  it  crosses  AD  prolonged ;  and 
so  on  for  each  required  side.  The  figure  Abcdef  is  the  one 
required. 


CONSTRUCTION  OF  EQUIVALENT  FIGURES. 

Prob.  58.  (PL  IV.  Fig.  59.)  To  construct  a  triangle  which 
shall  be  equivalent  to  a  given  parallelogram. 

Let  ABCD  be  the  given  parallelogram. 

1st.  Prolong  the  base  AB,  and  set  off  BE  equal  to  AB. 

2d.  Draw  lines  from  C,  to  A  and  E.  The  triangle  A  CE  is 
the  one  required. 

Prob.  59.  (PL  IV.  Fig.  60.)  To  construct  a  triangle  which 
shall  be  equivalent  to  a  given  quadrilateral. 

Let  ABCD  be  the  given  quadrilateral. 

1st.  Draw  a  diagonal,  as  AC. 

2d.  From  B  the  adjacent  angle  to  C,  to  which  the  diagonal 
is  drawn,  draw  a  line  parallel  to  A  C,  prolong  the  side  AB 
opposite  to  BCf  and  mark  the  point  F,  where  it  crosses  the 
parallel  to  AC. 


PROBLEMS   OF   EQUIVALENT   FIGURES,    &C.  (>3 

3d.  Draw  a  line  from  G  to  F.  The  triangle  FCD  is  the  one 
required. 

Prob.  60.  (PI.  IT.  Fig.  61.)  To  construct  a  triangle  equiva 
lent  to  any  given  polygon. 

Let  ABCDEFG  be  the  given  polygon. 

1st.  Take  any  side,  as  AB,  as  a  base,  and,  from  A  and  B, 
draw  the  diagonals  AF  and  BD  to  the  alternate  angles  to 
A  and  B. 

2d.  From  G  and  (7,  the  adjacent  angles,  draw  Oa  parallel 
to  FA,  and  Cb  to  DB. 

3d.  From  the  alternate  angles  F  and  Z>,  draw  the  lines 
Fa  and  Db.  A  figure  dbDEF  is  thus  formed,  which  is 
equivalent  to  the  given  one,  and  having  two  sides  less 
than  it. 

4th.  From  the  angles  a  and  &,  at  the  base  of  this  new 
figure,  draw  diagonals  to  the  alternate  angles,  to  a  and  b  (in 
the  Fig.  this  is  the  angle  E\  and  proceed,  precisely  as  in  the 
3d  operation,  to  form  another  figure  equivalent  to  the  last 
formed,  and  having  two  sides  less  than  it.  Proceed  in  this 
way  until  a  quadrilateral  or  pentagon  is  formed  equivalent 
to  the  given  figure,  and  convert  this  last  into  its  equivalent 
triangle,  which  will  be  the  one  required.  The  case  taken  for 
illustration  is  a  heptagon,  and  HEX  is  the  equivalent 
triangle. 

Prob.  61.  To  construct  a  triangle  equivalent  to  any  regular 
polygon. 

1st.  By  Prob.  49  find  the  radius  of  the  circle  inscribed  in 
the  polygon. 

2d.  Set  off  on  a  right  line  a  distance  equal  to  half  the  sum 
of  the  sides  of  the  polygon.  This  distance  will  be  the  base 
of  the  equivalent  triangle,  and  the  radius  of  the  inscribed 
circle  its  perpendicular  or  altitude. 


CONSTRUCTION  OF  CURVED  LINES  BY  POINTS. 

Prob.  62.  (Pl.IY.Fig.  62.)     To  construct  an  ellipse  on  given 
transverse  and  conjugate  diameters. 
Definitions.     An  ellipse  is  an  oval-shaped  curve.     The  line 


64  INDUSTRIAL   DRAWING. 

A — B  that  divides  it  into  two  equal  and  symmetrical  parts  is 
termed  the  transverse  axis.  The  line  C- — D,  perpendicular  to 
the  transverse  at  its  centre  point,  is  termed  the  conjugate  axis. 
The  points  A  and  B  are  termed  the  vertices  of  the  curve.  The 
points  E  and  F,  on  the  transverse  axis,  which  are  at  a  dis- 
tance from  the  points  G  and  D,  the  extremities  ef  the  con- 
jugate, equal  to  the  semi-transverse  0 — A,  are  termed  the 
foci  of  the  ellipse. 

The  ellipse  has  the  characteristic  feature  that  the  sum  of 
any  two  lines,  as  m — E  and  m — F,  drawn  from  a  point,  as  m, 
on  the  curve  to  the  foci,  is  equal  to  the  transverse  axis.  It  is 
this  characteristic  property  that  is  used  in  constructing  the 
curve  by  points. 

First  Method. 

Let  ab  be  the  length  of  the  transverse,  and  cd  that  of  the 
conjugate  diameter. 

1st.  Set  off  ab,  from  A  to  B,  on  any  line,  bisect  AB  by  a 
perpendicular,  and  set  off  on  this  perpendicular  the  equal 
distances  00,  and  OD,  each  equal  to  the  half  of  cd 

2d.  From  (7,  with  the  radius  OA,  describe  an  arc,  and 
mark  carefully  the  points  E  and  F,  where  it  crosses  AB. 

3d.  From  A,  take  off  any  distance  Ab,  and  mark  the 
point  b. 

4th.  With  the  distance  Ab  describe  an  arc  from  E,  and  a 
like  one  from  F. 

5th.  Take  off  the  remaining  portion  IB  of  AB ;  and  with 
it  describe  from  the  points  E  and  F  arcs,  and  mark  the  points 
m,  n,  o,  p,  where  these  arcs  cross.  These  are  four  points  of 
the  required  ellipse. 

6th.  To  obtain  other  points  of  the  curve  take  any  other 
point  on  AB,  as  c;  and  with  the  distances  Ac  and  cB, 
describe  arcs  from  E  and  F,  as  before.  The  points  where 
these  cross  are  four  more  points ;  and  so  on  for  as  many  aa 
may  be  required. 

Second  Method. 

Having  cut  a  narrow  strip  of  stiff  paper,  so  that  one  of  its 
edges  shall  be  a  straight  line,  mark  off  from  the  end  of  this 


CONSTRUCTION   OF  CURVED   LINES  BY    POINTS.          '65 

strip,  along  the  straight  edge,  a  distance  rt  equal  to  A  0,  half 
the  transverse  axis  of  the  ellipse ;  and  from  the  same  point  a 
distance  rs  equal  to  00,  half  the  conjugate  axis. 

1st.  Place  the  strip  thus  prepared  so  as  to  bring  the  point 
s  on  the  line  A B  of  the  transverse  axis,  and  the  point  t  on  the 
line  CD;  having  the  strip  in  this  position,  mark  on  the 
drawing  the  position  of  the  point  r;  this  is  one  point  of  the 
required  curve. 

2d.  Shift  the  strip  of  paper  to  a  new  position,  to  the  right 
or  left  of  the  first,  and  having  fixed  it  so  that  the  point  s  is 
on  AS,  and  the  point  t  on  CD,  mark  the  second  position  of 
the  point  r  ;  this  is  the  second  point  of  the  curve. 

By  placing  the  strip  so  that  the  first  point  marked  may  be 
near  A,  and  gradually  shifting  it  towards  C,  as  many  points 
may  be  marked  as  may  be  wanted ;  and  so  on  for  the 
remainder  of  the  curve. 

3d.  Through  the  points  thus  marked  draw  a  curved  line. 
It  will  be  the  required  ellipse. 

Remark.  The  accuracy  of  the  curve  when  completed  will 
depend  upon  the  steadiness  of  hand  and  correctness  of  eye  of 
the  draftsman.  When  the  points  of  the  curve  have  been 
obtained  by  the  first  method,  the  accuracy  of  their  position 
may  be  tested  as  follows  : — Joining  the  corresponding  points, 
as  m  and  n,  or  o  and  p,  above  and  below  the  line  AB,  by 
right  lines,  these  lines  mn  and  op  will  be  perpendicular  to 
AB,  and  be  bisected  by  it  if  the  construction  is  cor- 
rect. 

Prob.  63.  (PI.  V.  Fig.  63.)  Having  the  transverse  axis  of 
an  ellipse,  and  one  point  of  the  curve,  to  construct  the  conjugate 
axis. 

Let  AB  be  the  transverse  axis ;  and  a  the  given  point. 
1st.  Bisect  AB  by  a  perpendicular ;  and  from  the  centre 
point  0,  with  a  radius  OA,  describe  a  semicircle. 

2d.  Construct,  from  a,  a  perpendicular  to  AB,  and  mark 
wne  point  c  where  it  crosses  the  semicircle. 

3d.  Join  0  and  c ;  and  from  a  draw  a  parallel  to  AB,  and 
mark  the  point  r  where  the  parallel  crosses  Oc. 

4th.  From  0,  set  off  Or  to  C  on  the  perpendicular.     The 
distance  OC  is  the  required  semi-conjugate  axis. 
5 


66  INDUSTRIAL  DRAWING. 

Remark.  Having  the  semi-conjugate  other  points  of  the 
curve  can  be  found,  as  in  the  preceding  Prob. 

Prob.  64.  (PL  Y.  Fig.  63.)  At  a  point  on  the  curve  of  an 
ellipse  to  construct  a  tangent  to  the  curve. 

Let  m  be  the  point  at  which  the  tangent  is  to  be 
drawn. 

1st.  With  OA,  as  a  radius,  describe  a  semicircle  on  AB. 

2d.  From  m  construct  a  perpendicular  mq  to  AB,  and  mark 
the  point  n,  where  it  crosses  the  semicircle. 

3d.  At  n  construct  a  tangent  to  the  semicircle,  and  prolong 
it  to  cut  the  transverse  axis  prolonged  at  p. 

4th.  Through  p  and  m  draw  a  line.  This  is  the  required 
tangent. 

Prob.  65.  (PI.  V.  Fig.  63.)  From  a  point  without  an 
ellipse  to  construct  a  ta.ngent  to  the  curve. 

Let  D  be  the  given  point  from  which  the  tangent  is  to  be 
drawn. 

1st.  Join  the  point  D  with  0  the  centre  of  the  ellipse,  and 
mark  the  point  e  where  this  line  cuts  the  ellipse. 

2d.  From  0,  with  the  radius  OA,  describe  the  semicircle 
AhB. 

3d.  Through  e  draw  a  perpendicular  eq  to  the  transverse 
axis,  and  mark  the  point  g  where  it  cuts  the  semi- 
circle. 

4th.  Through  the  point  D  draw  a  perpendicular  Df  to  the 
transverse  axis,  and  prolong  it  towards  d. 

5th.  From  0  draw  a  line  through  g,  prolong  it  to  cut  the 
perpendicular  Df,  and  mark  the  point  d  of  intersection. 

6th.  From  d,  by  Prob.  22,  construct  a  tangent  to  the  semi- 
circle, and  mark  the  point  h  of  contact. 

7th.  From  h  draw  a  perpendicular  hk  to  the  transverse 
axis,  and  mark  the  point  i  where  it  cuts  the  ellipse. 

8th.  From  D  the  given  point  draw  a  line  through  i.  This 
is  the  required  tangent. 

Remark.  The  other  tangent  from  D  to  the  ellipse  can  be 
readily  obtained  by  constructing  the  second  tangent  to  the 
circle,  and  from  it  finding  the  point  on  the  ellipse  which 
corresponds  to  the  one  on  the  circle,  in  the  same  manner  a? 
the  point  i  is  found  from  h. 


CONSTRUCTION  OF  CURVED  LINES  BY  POINTS.  67 

Prob.  66.  (PL  IV.  Fig.  64.)  To  copy  a  given  curve  by 
points. 

Let  A  CD  be  the  given  curve  to  be  copied. 

1st.  Draw  any  line  as  AB  across  the  curve. 

2d.  Commencing  at  A  set  off  along  AB  any  number  of 
equal  distances  as  A~L,  1 — 2,  2 — 3,  &c. 

3d.  Through  the  points  1,  2,  3,  &c.,  construct  perpendicu- 
lars to  AB,  and  prolong  them  to  cut  the  curve  at  m,  n,  o,  p, 
&c. 

4th.  Having  drawn  a  right  line,  set  off  on  it  the  equal 
distances  A — 1,  1 — 2,  &c.,  taken  off  from  the  line  AB,  and 
through  the  points  thus  set  off  on  the  second  line  draw  per- 
pendiculars to  it.  From  the  points  where  these  perpendicu- 
lars cross  the  line,  commencing  at  the  first,  set  off  the 
distances  1 — m,  2 — o,  &c.,  on  the  portion  of  the  perpendicu- 
lars above  the  line ;  and  the  distances  1 — n,  2 — p,  &c.,  below 
it.  The  curve  drawn  through  the  points  thus  set  off  will  be 
a  copy  of  the  given  one ;  the  accuracy  of  the  copy  depending 
on  the  skill  of  the  draftsman. 

Prob.  67.  (PL  IV.  Figs.  65,  66.)  To  make  a  copy  of  a 
given  curve,  so  that  the  lines  of  the  copy  shall  be  greater  or  smaller 
than  the  corresponding  lines  of  tJie  given  curve  in  any  given  pro- 
portion. 

1st.  Having  drawn  a  line  AB  across  the  curve  (Fig.  64) 
set  off  along  it  the  equal  distances  A — 1,  1 — 2,  &c.,  and 
through  the  points  1,  2,  &c.,  construct  perpendiculars  to  AB, 
and  prolong  them  to  cut  the  curve  on  each  side  of  it. 

2d.  Draw  any  line,  as  ab  (Fig.  65),  on  which  set  off  equal 
distances  a — 1,  1 — 2,  &c.,  each  in  the  given  proportion,  take 
for  example  that  of  1  to  3,  to  those  set  off  on  the  given 
figure,  that  is,  make  a— 1  the  one-third  of  A — 1,  &c.,  and 
through  the  points  1,  2,  &c.,  construct  perpendiculars  to 
ib.  ' 

4th.  Set  off  any  given  line  cd  (Fig.  66),  with  the  distance 
ed  describe  two  arcs,  and  join  the  point  e,  where  they  cross, 
with  c,  and  d. 

5th.  From  e  set  off  cf  and  eg,  each  equal  to  one-third  of  cd, 
and  join /and  g. 

6th.  From  c  set  off  cm,  equal  to  1 — M  (Fig.  64) ;  co  equal 


08  INDUSTRIAL   DRAWING. 

2— -o,  &c. ;  join  e  with  the  points  m,  o,  &c. ;  and  mark  the 
points  r,  s,  &c.,  where  these  lines  cross  fg. 

7th.  Set  off  the  distance  fr  from  1  to  m  (F;g.  65) ;  fs  from 
2  to  o,  &<3.  The  points  m,  o,  &c.,  are  points  of  the  required 
copy.  In  like  manner  the  distances  n,  p,  &c.,  below  ab,  are 
constructed. 

Remark.  The  method  here  used  (Fig.  66)  for  constructing 
the  proportional  distances  fr,fs,  &c.,  to  those  cm,  co,  &c.,  can 
be  used  in  all  like  cases,  as  for  example  in  Prob.  17.  It 
furnishes  one  of  the  most  accurate  methods  for  such  cases,  as 
the  lines  drawn  from  c  cross  the  line  fg  so  as  to  mark  the 
points  of  crossing  r,  s,  &c.,  with  great  accuracy. 

Prob.  68.  (PL  Y.  Fig.  67.)  Through  three  given  points  to 
describe  an  arc  of  a  circle  by  points. 

Let  A,  B,  and  C  be  the  given  points. 

1st.  From  A,  with  the  radius  AC,  the  distance  between  the 
points  farthest  apart,  describe  an  arc  Co;  and  from  C  with 
the  same  radius  an  arc  Ap. 

2d.  From  A  and  C,  through  S,  draw  lines,  and  prolong 
them  to  a  and  b  on  the  arcs. 

3d.  From  b,  set  off  any  number  of  equal  arcs  b — 1,  1 — 2, 
&c.,  above  b;  and  from  a  the  same  number  of  equal  arcs 
below  a. 

4th.  From  (7,  draw  lines  C—  1,  C—  2,  &c.,  to  the  points 
above  b ;  and  from  A  lines  A — 1,  &c.,  to  the  corresponding 
points  below  a. 

5th.  Mark  the  points,  as  m,  &c.,  where  the  corresponding 
lines  A — 1  and  C— 1,  &c.,  cross.  These  are  points  of  the 
curve. 

6th.  Having  set  off  equal  arcs  below  5,  and  like  arcs  above 
a,  join  "the  corresponding  points  with  A  and  C.  The  points 
n,  &c.,  to  the  left  of  B,  are  points  of  the  required  curve. 

Remark.  This  construction  is  only  useful  when,  from  the 
position  of  the  given  points,  the  centre  of  the  circle  which 
would  pass  through  them  cannot  be  constructed. 

Prob.  69.  (PL  Y.  Fig.  68.)  Having  given  the  axis,  the 
vertex,  and  a  point  of  a  parabola,  to  find  other  points  of  the 
curve  and  describe  it. 


PROBLEMS   OF   CIRCLES,   &C.  (Jf) 

Let  AS  be  the  axis ;  A  the  vertex ;  and  0  the  given 
point. 

1st.  From  C  draw  a  perpendicular  to  AB,  and  mark  the 
point  d  where  it  crosses  AS. 

2d.  At  A  construct  a  perpendicular  to  AS,  and  from  C  a 
parallel  to  AS,  and  mark  the  point  F  where  these  two  lines 


3d.  Divide  Cd  and  CF  respectively  into  the  same  number 
of  equal  parts,  say  four  for  example. 

4th.  From  the  points  of  division  1,  2,  3,  on  Cd  draw 
parallels  to  AS ;  and  from  the  point  A  lines  to  the  points 
1,  2,  3  on  CF. 

5th.  Mark  the  points  x,  y,  z,  where  the  lines  from  A  cross 
the  corresponding  parallels  to  AS.  These  will  be  the  required 
points  through  which  the  curve  is  traced. 

6th.  Through  the  points  x,  y,  z,  drawing  perpendiculars  to 
AS,  and  from  the  points  a,  b,  c,  where  they  cross  it,  setting 
off  distances  oo/,  by',  and  czf  respectively  equal  to  ax,  &c. ; 
the  points  a/,  ?/,  zf  will  be  the  portion  AD  of  the  curve  below 
AS. 

Prob.  70.  (PL  V.  Fig.  69.)  Having  given  the  diameter  of 
a  circle,  to  construct  a  right  line  which  shall  be  equal  in  length  to 
its  circumference. 

Let  AM  be  the  given  diameter. 

1st.  Draw  a  right  line,  and  having,  from  any  convenient 
scale  of  equal  parts,  taken  off  a  distance  greater  than  the 
given  diameter,  and  equal  to  113  of  these  equal  parts,  set  it 
off  from  a  to  b. 

2d.  From  a  and  b,  with  the  distance  ab,  describe  arcs,  and 
from  the  point  c,  where  they  cross  each  other,  draw  lines 
through  a  and  b. 

3d.  Take  off  from  the  scale  a  distance  equal  to  355  equal 
parts,  and  set  it  off  from  c,  to  dt  and  e,  on  the  lines  drawn 
through  a  and  b  ;  and  join  the  points  d  and  e. 

4th.  From  or,  set  off  on  ab,  the  distance  am,  equal  to  the 
given  diameter  AM;  and  from  c,  draw  a  line  through  m,  and 
prolong  it  to  cross  de  at  n.  The  distance  dn  is  the  required 
length  of  the  circumference. 


70  INDITSTBIAL 


CHAPTEK  IV. 


TINTING    AND    SHADING. 

Line  Shading. 

1.  Flat  tints.  It  is  often  necessary  to  cover  a  surface  with 
equi-distant  parallel  lines.  This  is  good  practice  for  the 
eye,  as  well  as  the  hand,  as  the  distance  between  the  lines 
should  be  obtained  by  the  eye  alone ;  use  the  triangles  for 
this,  sliding  one  along  the  other  or,  better,  against  the  square. 

When  the  spacing  is  uniform  and  the  lines  smooth,  it  gives 
the  effect  of  a  flat  tint.  If,  while  doing  this,  you  should  hap- 
pen to  make  a  space  larger,  or  smaller,  than  the  preceding, 
do  not  make  the  next  line  at  the  regular  distance  from  the 
last,  as  this  would  make  the  break  in  the  spacing  more  notice- 
able, but  gradually  reduce,  or  increase  this  distance  until  the 
regular  space  is  reached,  and  then  continue  with  that. 

These  irregularities  in  spacing  are  less  noticeable  where 
the  spacing  is  coarse ;  so  that  it  is  well  in  beginning  to  make 
the  lines  at  least  ^  of  an  inch  apart.  With  practice  this  can 
be  reduced ;  the  fineness  of  the  spacing  should  have  some 
reference  to  the  size  of  the  figure. 

This  kind  of  shading  is  used  mostly  for  sections,  as  shown 
in  Fig.  156.  PL  XVII. ;  it  being  customary  to  run  the  lines 
45°  in  either  direction.  Where  the  sections  of  two  bodies 
join,  as  in  Fig.  3.  PL  I.*  let  the  lines  run  in  opposite  direc- 
tions ;  where  there  are  more  than  two  bodies,  try  to  arrange 
so  that  on  no  two  adjacent  ones  the  lines  run  in  the  same 
direction  ;  if  necessary  change  the  angle. 

For  practice  take  four  rectangles  and  cover  with  lines,  aa 
shown  in  Fig.  2.  PL  I*.  Afterwards  try  Figs.  3,  4,  5,  using 
finer  spacing.  The  student  can  easily  vary  these  figures. 


TINTING   AND   SHADING.  71 

2.  Graduated  tints.  There  are  two  methods  by  which  a 
graduated  tint  may  be  obtained  ;  first,  by  varying  the  distances 
without  changing  the  size  of  the  line  (PI.  I*.  Fig.  6),  or  sec- 
ond, by  changing  the  size  of  the  lines  as  well  as  the  distances 
(PL  I.*  Fig.  7).  The  effect  of  the  last  method  is  the  best ; 
always  shade  from  the  dark  line  to  the  li^ht.  Try  shading  a 
rectangle  by  each  method. 

For  further  practice  try  the  shading  of  an  hexagonal  prism 
and  cylinder ;  as  seen  in  Figs.  8  and  9,  PI.  I.*,  the  shading 
on  these  surfaces  is  made  up  of  flat  and  graduated  tints. 
When  shading  the  cylinder  make  the  darkest  line  first,  and 
shade  both  ways  from  it ;  the  shade  on  the  left-hand  side 
should  be  made  from  left  to  right. 

Figs.  10  and  11,  PI.  I.*  give  practice  with  the  compasses 
in  producing  graduated  tints.  In  the  first  (fig.  10),  each 
circle  is  to  be  completed  with  a  uniform  line,  the  shade  being 
lightest  towards  the  centre ;  in  the  second  (fig.  11),  each  circle 
is  made  with  tapering  lines. 

It  requires  some  practice  to  make  a  tapering  line  with  the 
compasses ;  when  the  circle  is  to  be  complete,  set  the  pen  to 
the  size  of  the  finest  part  and  describe  the  circle;  then 
separate  the  points  of  the  pen  a  little,  and  sweep  over  the 
heaviest  part  of  the  circle ;  if  the  pen  is  brought  in  contact 
with  the  paper,  and  also  taken  from  it  while  it  is  being  turned, 
and  the  pressure  upon  the  paper  is  varied,  making  it  greatest 
at  the  darkest  part  of  the  circle,  a  very  good  taper  can  be 
given  to  the  line.  It  would  be  well  to  protect  the  centre  in 
making  these  figures. 

The  rules  for  locating  the  dark  and  light  parts  on  the  solids 
just  mentioned,  are  given  in  the  chapter  on  shading;  they 
are  introduced  here  merely  for  practice,  and  the  examples 
given  will  be  sufficient  guides. 

India  Ink  Shading. 

3,  Flat  tints.  Tinting  with  India  ink  is  a  quicker  and 
easier  method  of  shading  than  by  the  use  of  lines. 

For  tinting  one  needs  to  have  at  hand  a  tumbler  of  clean 
water  and  two  or  three  brushes  of  different  sizes ;  those  with 


72  INDUSTRIAL   DBAWING. 

large  bodies  and  fine  points  are  best,  as  they  will  hold  consid- 
erable tint.  To  prepare  the  tint,  rub  the  cake  of  ink  upou 
the  tile  and  then  take  from  that  with  a  brush,  and  mix  with 
the  water  in  the- tumbler,  until  the  desired  shade  is  obtained. 
Care  should  be  taken  that  the  brush  and  tumbler  are  perfectly 
clean,  and  it  is  well  to  keep  the  tumbler  covered  after  the  tint 
is  prepared.  Do  'not  make  the  tint  as  dark  as  you  wish  it 
upon  the  drawing,  when  finished ;  it  is  much  easier  to  lay  a 
light  tint  smoothly,  than  a  dark  one,  so  that  it  is  better  to  get 
the  depth  of  shade  required  by  successive  washes ;  let  each 
wash  dry  before  laying  another. 

As  a  rule,  it  is  better  to  go  over  the  surface  to  be  tinted 
first  with  clean  water,  as  the  first  wash  will  lay  smoother ;  and 
if  the  first  wash  is  spotted,  it  will  show  through  all  the  rest ; 
this  damping  is  especially  necessary  if  the  tint  used  is  dark, 
or  the  surface  large;  when  the  surface  is  small,  and  some 
skill  has  been  acquired,  the  damping  may  be  omitted. 

Do  not  use  India  rubber  upon  the  surface  to  be  tinted,  as 
it  is  difficult,  to  lay  a  smooth  tint  afterwards ;  avoid  also  rest- 
ing the  hands  upon  the  surface,  as  any  moisture  from  them 
will  affect  the  flow  of  the  tint. 

For  practice  try  laying  flat  tints  upon  rectangles  of  differ- 
ent sizes,  commencing  with  small  ones.  When  doing  this, 
commence  at  the  top  and  work  down,  keeping  the  advancing 
edge  nearly  horizontal  and  always  wet ;  let  the  board  be  in- 
clined a,  little,  so  that  the  tint  will  follow  the  brush ;  upon 
reaching  the  bottom  of  the  rectangle,  if  there  is  any  surplus 
tint  upon  the  paper,  it  should  be  removed  with  the  brush, 
haying  first  wiped  it  dry. 

Try  next  some  surface  where  it  is  necessary  to  watch  two 
or  more  edges,  to  see  that  they  do  not  get  dry ;  the  space  be- 
tween two  concentric  circles  will  do,  or  trace  the  outlines  of 
the  irregular  curve  upon  paper,  and  either  tint  the  curve  or 
else  tint  around  the  curve,  leaving  that  white.  Let  the  outlines 
of  surfaces  to  be  tinted  be  fine  hard  pencil  lines ;  with  care 
in  laying  the  tints,  they  make  the  best  finish  for  the  edge,  but 
where  the  edge  has  become  uneven,  a  fine  light  line  of  ink 
will  improve  it. 

4.  Graduated  tints.     Having  acquired  some  skill  in  laying 


TINTING   AND   SHADING.  73 

flat  tints,  try  next  a  graduated  tint.  There  are  several  methods 
of  doing  this,  one  of  which  is  by 

Flat  tints.  When  it  is  desired  to  tint  a  rectangle  so  that 
it  shall  be  darkest  at  the  top,  and  shade  off  lighter  towards 
the  bottom,  divide  the  sides  into  any  number  of  equal  parts 
with  pencil  marks  (PI.  I.*  Fig.  12.);  commencing  at  the  top, 
lay  a  flat  tint  upon  the  first  space ;  when  this  is  dry,  commence 
at  the  top  and  lay  a  flat  tint  over  the  first  two  spaces.  Pro- 
ceed in  this  way,  commencing  at  the  top  each  time,  until  the 
whole  rectangle  is  covered ;  by  making  the  divisions  of  the 
rectangle  quite  small,  the  effect  is  more  pleasing.  See  that 
the  lower  edges  of  the  flat  tints  are  as  straight  as  possible,  for 
they  show  through  the  succeeding  tints,  and  detract  much  from 
the  appearance  when  irregular ;  it  is  not  well  to  make  the  divi- 
sion marks  across  the  surface,  as  they  would  be  likely  to  show. 

There  must  be  sufficient  time  between  the  tints  to  allow 
the  top  space  to  dry,  else  some  of  its  tint  will  be  washed  to 
the  spaces  below,  and  when  completed,  the  upper  spaces  of 
the  rectangle  will  have  the  same  tint. 

It  is  better  to  follow  the  order  given,  instead  of  going  over 
the  whole  surface  for  the  first  tint,  and  reducing  the  surface, 
by  one  space,  for  each  succeeding  tint ;  by  the  first  method 
the  edges  are  covered  by  the  over-laying  tints,  and  a  softer 
appearance  is  given  than  would  be  obtained  by  the  second 
method.  Figs.  11, 12,  PI.  I.**,  give  examples  of  shading  with 
flat  tints. 

5.  Softened  tints.     This  gives  a  much  smoother  appearance 
than  the  last  method.    Divide  the  rectangle  as  before,  and 
tint  the  first  space ;  but  instead  of  letting  the  edge  dry.  as  a 
line,  wash  it  out  with  clean  water ;  wash  out  the  edge  of  each 
tint  in  the  same  way;  and  when  finished,  there  will  not  be 
any  abrupt  changes  from  one  tint  to  another,  as  in  the  last 
method. 

6.  Dry  shading.     First  tint  the  rectangle  by  the  method 
of  flat  tints ;  then  keeping  the  brush  pretty  dry,  so  that  the 
strokes  disappear  a.bout  as  fast  as  made,  use  it  as  you  would 
a  pencil,  and  shade  between  the  edges  until  they  disappear ; 
keep  the  point  of  the  brush  pretty  fine,  and  make  the  strokes 
short  and  parallel  to  the  edges. 


74:  INDTTSTRTAT.  DBA  WING. 

This  takes  considerable  time,  but  gives  a  very  good  result ; 
not  as  smooth  as  by  softened  tints,  but  full  as  pleasant  to  the 
eye. 

When  the  surface  to  be  shaded  is  quite  small,  the  flat  tints 
might  be  omitted,  and  the  tint  applied  with  short  strokes  of 
the  brush,  wherever  needed. 

For  practice,  try  the  different  methods  upon  rectangles 
about  2  by  4  inches.  Let  the  shade  be  darkest  at  the  top,  or 
bottom ;  let  the  shade  be  darkest  along  the  right  or  left  hand 
edges ;  let  it  be  darkest  through  the  centre,  either  horizontal- 
ly or  vertically ;  in  the  last  case  the  centre  space  is  the  first 
one  tinted,  and  the  second  wash  covers  the  centre  and  two 
adjacent  spaces. 

7.  Colors.  The  directions  previously  given  will  apply  to 
the  use  of  colors  in  forming  flat  or  graduated  tints ;  another 
method  may  be  used  for  graduated  tints,  and  that  is  to  shade 
the  surface  first  with  India  ink,  and  then  cover  it  with  a  flat 
tint  of  the  color. 


REPRESENTING   DIFFERENT  MATERIALS.  75 


CHAPTER  Y. 

CONVENTIONAL  MODES   OF  REPRESENTING   DIFFERENT  MATERIALS. 

1.  In  mechanical  drawing  different  materials  may  be  repre- 
sented by  appropriate  conventional  coloring,  which  is  usually 
done  in  finished  drawings;  or  else  by  simply  drawing  the 
outlines  of  the  parts  in  projection,  and  drawing  lines  across 
them  according  to  some  rule  agreed  upon,  to  represent  stone, 
wood,  iron,  etc.     Although  there  is  no  uniformity  in  the  use 
of  conventional  tints,  the  following  will  be  found  convenient 
for  this  purpose : 

Cast  iron-,  Payne's  grey. 

Wrought  iron,  Prussian  blue. 

Steel,  Prussian  blue  and  carmine. 

Brass,  Gamboge. 

Copper,  Gamboge  and  carmine. 

Stone,  Sepia  and  yellow  ochre. 

Brick,  Light  red. 

Wood,  Burnt  Sienna. 

Earth,  Burnt  Umber. 

For  stone  a  light  tint  of  India  ink  might  be  used  instead  of 
the  colors  given  above ;  if  a  little  carmine  be  added  to  the 
light  red,  when  used  to  represent  brick,  it  will  make  a 
brighter  color;  raw  sienna  might  be  used  instead  of  burnt 
sienna  for  wood. 

2.  Wood.     When  the  drawing  is  on  a  small  scale,  the  out- 
,  lines  only  of  a  beam  of  timber  are  drawn  in  projection,  when 
'  the  sides  are  the  parts  projected,  PI.  VII.  Fig.  88 ;  when  the 

end  is  the  part  projected,  the  two  diagonals  of  the  figure  are 
drawn  on  the  projection. 

If  a  longitudinal  section  of  the  beam  is  to  be  shown,  PL 
VII.  Fig.  89,  fine  parallel  lines  are  drawn  lengthwise.  In  a 
cross  section,  fine  parallel  lines  diagonally. 


76  INDUSTRIAL   DRAWING. 

Where  the  scale  is  sufficiently  large  to  admit  of  some 
resemblance  to  the  actual  appearance  of  the  object  being 
attempted,  lines  may  be  drawn  on  the  projection  of  the  side 
of  the  beam,  PI.  VII.  Fig.  90,  to  represent  the  appearance  of 
the  fibres  of  the  wood ;  and  the  same  on  the  ends. 

In  longitudinal  sections  the  appearance  of  the  fibres  may 
be  expressed,  PI.  VII.  Fig.  91,  with  fine  parallel  lines  drawn 
over  them  lengthwise.  In  cross  sections  the  grain  may  be 
shown,  as  in  the  projection  of  the  ends,  with  fine  parallel  lines 
diagonally. 

Figs.  13, 14, 15,  PI.  I.*,  give  additional  examples  of  the 
method  of  representing  wood  in  line  drawings.  Before  at- 
tempting to  represent  graining,  it  would  be  well  to  examine 
the  graining  of  wood  in  the  material  itself,  and  to  have  a 
piece  at  hand  to  imitate ;  in  the  floor  of  the  drawing-room 
may  often  be  found  good  examples. 

In  beginning  to  grain  a  timber,  make  the  knots  first,  and 
then  fill  in  the  remaining  space  with  lines,  arranging  so  as  to 
enclose  the  knots ;  until  some  practice  has  been  acquired,  it 
would  be  well  to  pencil  the  graining  before  inking.  The  size 
of  the  knots  should  have  some  reference  to  the  size  of  the  tim- 
ber ;  let  the  graining  lines  be  made  with  a  tine  steel  pen,  using 
light  ink ;  where  the  wood  is  to  be  colored,  apply  the  tint  be- 
fore graining. 

"When  the  timber  is  large,  use  a  brush  to  make  the  grain- 
ing ;  if  it  is  to  be  colored,  use  a  dark  tint  of  burnt  sienna  for 
the  lines,  and  wash  over  with  a  lighter  tint  of  the  same ;  when 
using  the  brush  for  graining,  make  the  points  of  the  knots 
widest,  as  in  Fig.  15,  PI.  L* ;  it  would  add  also  to  the  looks 
of  graining  made  with  a  pen  to  widen  the  ends  of  the  knots 
with  the  brush. 

Fig.  14,  PI.  I.*,  differs  from  Fig.  13  only  in  having  a  series 
of  short  marks  introduced  in  the  graining ;  if  desired  to  dis- 
tinguish between  soft  and  hard  wood,  let  Fig.  13  represent* 
soft,  and  Fig.  14  hard  wood. 

In  Fig.  15,  PI.  I.*,  the  graining  is  made  up  of  a  series  of 
short  hatches ;  when  done  nicely,  this  gives  a  very  good  effect. 
This  style  may  also  be  used  to  represent  hard  wood. 

The  cross  sections,  Figs.  13,  14,  15,  PI.  I.*,  are  shown  by  a 


REPRESENTING  DIFFERENT   MATERIALS.  77 

series  of  concentric  circles,  with  a  few  lines  radiating  from 
the  centre,  representing  cracks;  use  the  bow  compasses  to 
describe  the  circles,  from  a  centre  either  within  or  without 
the  section.  When  the  section  is  narrow,  as  at  <z,  Fig.  13, 
PL  I.*,  parallel  straight  lines  are  sufficient. 

To  represent  the  end  of  the  timber  as  broken,  let  the  line 
be  made  up  of  a  series  of  sharp  points.  Where  a  number  of 
adjoining  planks  are  represented  as  broken  along  nearly  the 
same  line,  let  the  end  of  each  be  distinct,  as  in  Fig.  1,  PI.  I.* 

3.  Masonry.  In  drawings  on  a  small  scale,  the  outlines  of 
the  principal  parts  are  alone  put  down  on  the  projections ; 
and  on  the  parts  cut,  fine  parallel  wavy  lines,  drawn  either 
vertically  or  horizontally  across,  are  put  in. 

In  drawings  to  a  scale  sufficiently  large  to  exhibit  the  de- 
tails of  the  parts,  lines  may  be  drawn  on  the  elevations,  PL 
VII.  Fig.  92,  to  show  either  the  general  character  of  the  com- 
bination of  the  parts,  or  else  the  outline  of  each  part  in  de- 
tail, as  the  case  may  require.  In  like  manner  in  section,  the 
outline  of  each  part,  PL  VII.  Fig.  93,  in  detail,  or  else  lines 
showing  the  general  arrangement,  may  be  drawn,  and  over 
these  fine  parallel  lines. 

Figs.  16-23,  PL  I.*,  give  additional  examples  of  stone  work. 
In  Figs.  16,  17  there  is  a  ledge  cut  around  each  stone,  and 
sunk  below  the  face.  The  face  may  be  dressed  with  a  pick, 
Fig.  17,  or  it  might  be  left  rough,  and  represented  as  in  Figs. 
21-23,  PL  I.*.  Fig.  16  would  be  the  method  of  representing 
such  stone  work  in  a  line  drawing ;  the  lower  and  right-hand 
lines  of  the  inner  rectangle  should  be  heavy. 

Fig.  18,  PL  I.*,  represents  the  surface  of  the  stone  a8 
finished ;  where  a  number  of  stones  come  together,  leave  a 
narrow  line  of  light  at  the  top  and  left-hand  edges  of  each. 

In  Figs,  19,  20,  PL  I.*,  the  edges  of  the  stones  are  bevelled, 
and  the  surfaces  finished.  Fig.  19  is  the  method  of  repre- 
senting in  a  line  drawing.  For  a  finished  shaded  drawing, 
Fig.  20  would  do,  shaded  with  the  stone  tint  alone,  or  with 
India  ink  first,  and  then  a  flat  stone  tint ;  the  depth  of  shade 
upon  each  face  of  the  stone  is  shown  in  the  example 
given. 

Figs.  21-23,  PL  I.*,  give  different  methods  of  representing 


78  INDUSTRIAL  DRAWING. 

stone  where  the  .joints  only  are  dressed,  and  the  faces  ]eft 
rough. 

4.  Metals.     The  conventional  colors  for  the  different  metals 
have  already  been  given. 

To  represent  metal  as  broken,  the  line  should  be  somewhat 
irregular,  but  not  as  much  so .  as  for  the  breaking  of  wood ; 
Fig.  1'.  PI.  I.*  gives  examples  of  the  customary  method  of 
showing  the  broken  ends  of  rods,  bars,  etc. ;  the  parallel  lines 
may  be  either  black  or  the  conventional  color. 

5.  Earth.     In  vertical  sections  extending  below  the  level  of 
the  ground  it  may  be  necessary  to  show  the  section  ^of  the 
earth,  etc.,  around  the  foundations.     If  the  soil  is  common 
earth  this  may  be  done,  as  shown  in  PI.  VIII.  Fig.  94.     If 
sand,  as  in  PI.  VIII.  Fig.  95.     If  stony,  as  in  PI.  VIII.  Fig. 
96.     If  solid  rock,  as  in  PI.  VIII.  Fig.  97.     Where  the  earth 
is  embanked  its  section  may  be  shown,  as  in  PI.  VIII.  Fig.  98. 

If  it  is  desired  to  color  the  sections  of  earth,  let  a  flat  tint 
of  burnt  umber  be  laid  first,  the  lower  edge  being  washed 
out,  then  when  dry  the  lines  may  be  drawn  over  it  with  a 
pen  ;  or,  after  laying  the  flat  tint  of  burnt  umber,  with  the 
brush,  make  a  series  of  short  horizontal  strokes  over  the  wash 
already  laid,  varying  the  shade  of  the  tint  for  the  strokes, 
some  being  quite  dark. 

6.  Water.     This  is  represented  as  in  Fig.  24.  PL  I.*  by 
horizontal  lines ;  for  colored  drawings  make  the  lines  blue. 

Fig.  24.  PI.  1.*  represents  a  section  of  a  canal ;  for  practice 
let  this  be  drawn  and  finished  in  colors. 


OONSTBUCTION   OF   EEGULAE   FIGURES.  79 


CHAPTER  VL 

CONSTBUOTION   OF   EEGULAE   FIGUEES. 

1.  The  following  figures  are  valuable  as  exercises  in  con- 
struction.    Simple  as  they  may  seem,  one  trial  will  convince 
that  care  is  necessary  in  every  step  to  secure  a  good  result ; 
let  the  construction  lines  be  fine  and  light  and  the  intersec- 
tions good.     These  figures  afford  further  practice  for  tinting 
both  with  lines  and  colors ;  use  different  shades  of  the  same 
color,  or  different  colors,  laying  the  lightest  first.    The  arrange- 
ment indicated  upon  the  figures  may  be  followed,  or  changed 
to  suit  the  taste. 

2.  To  represent  a  pavement  made  up  of  squares.    (PI.  I.** 
Fig.  1.)     Divide  the  lower  side  into  any  number  of  equal 
parts,  and  through  the  points  of  division  draw  lines  45°  both 
ways  ;  if  through  the  points  where  these  diagonals  meet  the 
sides,  diagonal  lines  be  drawn,  they  will  complete  the  figure. 

3.  To  represent  a  pavement  of  equilateral  triangles.     (PI. 
I.**  Fig.  2.)    Divide  the  lower  side  into  any  number  of  equal 
parts,  and  through  these  points  draw  lines  60°  each  way ; 
through  the  points  in  which  these  intersect,  draw  horizontal 
lines. 

4.  To  represent  a  pavement  of  hexagons.     (PI.  I.**  Fig.  3.) 
Having  given  the  side  of  the  hexagon,  lay  it  off  any  number 
of  times  upon  the  side,  and  draw  lines  60°  through  these 
points;  the  method  of  completing  the  figure  will  then  be 
apparent ;  a  c  is  equal  to  half  the  side  of  the  hexagon.     . 

5.  To  represent  a  pavement  made  up  of  octagons  and 
squares.    (PI.  I.**  Fig.  4.)    Divide  the  surface  first  into 
squares  ;  then  construct  an  octagon  in  one  of  these  ;  if  it  is 
desired  to  have  a  regular  octagon,  it  may  be  done  by  taking 
half  of  the  diagonal  of  the  square  as  a  radius  and  describing 
arcs  from  each  corner  of  the  square  as  a  centre,  until  they  in- 


80  INDUSTRIAL   DRAWING. 

tersect  the  sides ;  this  is  indicated  in  one  of  the  squares  of  the 
figure.  After  constructing  one  octagon,  the  others  may  be 
obtained  from  it  by  projecting,  as  shown  by  the  dotted  lines. 

6.  To  represent  a  pavement  made  up  of  isosceles  triangles. 
(PI.  I.**  Fig.  5.)     Divide  the  lower  side  into  any  number  of 
equal  parts,  and  through  the  points  of  division  draw  vertical 
lines ;  through  every  other  point  of  division  draw  lines  at  30° 
and  60° ;  draw  horizontal  lines  through  the  points  in  which 
the  lines  at  60°  intersect.     If  the  lines  at  60°  are  left  out',  the 
surface  will  be  covered  with  rhombuses. 

7.  In  Fig.  6.  PI.  I.**,  the  surface  is  covered  with  intersect- 
ing and  tangent  circles.     First  divide  into  squares,  as  shown 
by  the  dotted  lines ;  then  describe  the  large  circles,  and  next 
the  small  ones  tangent. 

8.  Figs.  7-10.  PI.  I.**,  are  examples  of  architectural  orna- 
ment.    The.  method  of  constructing  will  be  evident  from  the 
figures.    In  Fig.  7,  for  a  change,  the  dark  part  might  be  left 
white  and  the  rest  shaded. 


PEOJECTION8 


CHAPTER  YIL 

PROJECTIONS. 

BY  the  methods  given  in  the  preceding  problems,  we  aro 
enabled  to  construct  most  of  the  geometrical  figures  that  can 
be  traced,  or  drawn  on  a  plane  surface,  according  to  geometri- 
cal principles,  and  with  the  ordinary  mathematical  instru- 
ments ;  and  if,  in  a  practical  point  of  view,  our  object  was 
simply  to  obtain  the  shape  and  dimensions  of  such  forms  as 
could  be  cut  from  a  sheet  of  paper,  a  thin  board,  or  a  block 
of  wood  of  uniform  thicknessx  these  problems  would  be  suffi- 
cient for  the  purpose  ;  for  it  would  be  only  necessary  to  con- 
struct the  required  figure  on  one  side,  or  end  of  the  board,  or 
block,  and  then  cut  away  the  other  parts  exterior  to  the  out- 
line of  the  figure.  Here  then  \ve  have  an  example,  in  which 
the  form  and  dimensions  of  one  side,  or  end  of  a  body,  being 
given,  the  body  itself  can  be  shaped  by  means  of  this  one 
view ;  and  this  is  applicable  to  all  cases  where  the  thickness 
of  the  body  is  the  same  throughout,  and  where  the  opposite 
sides,  or  ends,  are-figures  precisely  alike  in  shape  and  dimen- 
sions. This  method  is  applicable  to  a  numerous  class  of  bodies 
to  be  met  with  in  the  industrial  arts ;  particular  cases  will 
readily  occur  to  any  one.  Among  the  most  simple,  the  com- 
mon brick  may  be  taken  as  an  example ;  the  thickness  in  this 
case  is  uniform,  and  the  opposite  sides  are  equal  rectangles ; 
hence  taking  a  board  of  the  same  thickness  as  the  brick,  mark- 
ing out  on  its  surface  the  rectangle  of  the  side,  and  then  cutting 
away  the  portion  of  the  board  exterior  to  this  outline,  a  solid 
will  be  obtained  of  the  same  form  and  dimensions  as  the  brink. 

But  it  is  evident  that  a  drawing  of  the  rectangle  that  repre- 
sents the  side  would  not  be  sufficient  to  determine  the  form 
of  the  entire  brick,  if  we  did  not  know  its  thickness.  In  order 
that  the  drawing  shall  represent  the  forms  and  dimensions  of 
all  the  faces  of  the  brick,  it  is  obvious  that  some  means  must 
be  resorted  to  by  which  these  parts  can  also  be  represented. 
6 


82 


INDUSTRIAL  DBA. WING. 


The  necessity  for  this  will  be  still  more  apparent  when  we 
desire  to  represent  the  forms  and  dimensions  of  bodies,  which, 
although  of  uniform  thickness,  have  their  opposite  faces  of 
different  forms  and  dimensions ;  and  more  especially  in  the 
more  complicated  cases,  where  the  surface  of  the  body  is 
formed  of  figures  differing  both  in  forms  and  dimensions 
from  each  other.  The  means  by  which  we  effect  this  is 
termed  the  method  of  projections.  By  it  we  are  enabled  to 
represent  the  forms  and  dimensions  of  all  the  parts  of  a  body, 
however  complicated,  provided  they  can  be  constructed  by 
geometrical  principles. 

Principles. 

If  through  a  given  point,  A  (Fig.  19),  in  space  a  line  be 
drawn  perpendicular  to  any  plane,  M  JV,  the  point  a  in  which 


FIG.  19. 

the  line  pierces  the  plane  is  called  the  projection  of  the  point 
A  upon  that  plane. 

The  projection  of  a  right  line  is  obtained  by  joining  the 
projections  of  any  two  of  its  points;  for  example,  given  the 


line  A  B  (Fig.  20),  in  space,  a  is  the  projection  of  A,  and  b  of 
B ;  joining  a  b  we  have  the  projection  of  A  B. 

If  the  given  line  be  curved  instead  of  straight,  we  have  to 
find  the  projections  of  more  of  its  points  and  join  them ;  the 


PEOJEOTIONS. 


83 


more  points  found  the  more  accurate  the  projection.  Fig.  21 
illustrates  this,  where  a  h,  the  projection  of  A  H,  is  found  by 
joining  the  projections  of  its  different  points. 


A  solid  being  given  to  find  its  projection,  it  will  be  ob- 
tained by  joining  the  projections  of  its  bounding  edges. 

It  is  evident  in  these  examples  given,  that  it  is  impossible 
to  determine  from  the  projections  alone,  the  distance  of  the 
points  in  space  from  the  plane ;  in  other  words,  the  position 
of  the  points  in  space  is  not  fixed. 

Having  seen  that  one  projection  is  not  sufficient  to  locate  a 
point  in  space,  and  as  there  can  be  only  one  projection  of  a 
point  on  any  plane,  suppose  we  take  two  planes  at  right  angles 
to  each  other,  and  find  the  projections  of  a  point  upon  each 
of  these. 


Fig.  22  shows  pictorially  the  two  planes  G  N  and  O  P  at 
right  angles ;  A  is  a  point  in  space,  a  and  a'  are  its  projec- 
tions on  either  plane ;  the  distance  of  A  from  the  plane  G  Nt 


84  INDUSTRIAL   DRAWING. 

or  A  a,  is  equal  to  the  distance  a\  and  the  distance  of  A  from 
the  plane  G  P,  or  A  a',  is  equal  to  a  t>.  If  at  the  points  a,  and 
of  perpendiculars  should  be  erected  to  each  plane,  we  see  that 
these  must  intersect  at  the  point  A  in  space ;  and  as  these 
perpendiculars  can  intersect  in  only  one  point,  it  follows  that 
there  is  only  one  point  in  space  that  can  be  projected  in  a 
and  a'. 

Thus  we  see,  if  the  projections  of  a,  point  are  found  upon 
two  planes  at  right  angles,  its  position  in  space  is  fixed,  and 
can  be  determined  from  the  projections. 

If  the  projections  of  two  or  more  points  are  found  upon 
these  same  planes,  we  should  not  only  be  able  to  determine 
their  positions  respecting  the  planes,  but  also  their  relative 
position  ;  hence,  it  follows,  if  a  solid  be  projected  upon  these 
two  planes,  we  can  determine  its  dimensions  from  the  projec- 
tions. 

We  have  found,  then,  that  two  planes,  at  right  angles,  are 
necessary  in  projections;  these  are  called,  respectively,  the 
horizontal  and  vertical  planes  of  projection. 

The  line  of  intersection,  G  L  (Fig.  22),  is  called  the  ground 
line.  The  point  a  is  called  the  horizontal  projection  of  A, 
and  a'  the  vertical  projection.  The  horizontal  projection  of 
an  object  is  often  called  the  plan,  and  the  vertical  projection 
the  elevation. 

The  perpendiculars  through  A  are  called  the  projecting 
lines ;  the  plane  of  the  two  is  perpendicular  to  both  planes  of 
projection,  and  also  the  ground  line,  and  intersects  both  planes 
in  right  lines,  perpendicular  to  the  ground  line  at  the.  same 
point. 

Since  it  is  impracticable  to  draw  upon  two  planes  at  right 
angles,  the  vertical  plane  is  considered  as  revolved  back,  about 
the  ground  line,  until  it  forms  one  and  the  same  surface  with 
the  horizontal  plane.  By  this  revolution  the  relative  position 
of  points  in  the  vertical  plane  is  not  affected ;  every  point 
remains  at  the  same  distance  from  the  ground  line  after  revo- 
lution as  before.  This  is  shown  in  Fig.  22,  where  the  vertical 
plane  G  P  is  represented  as  revolved  back  to  the  horizontal 
position  Cr  P' ;  a'  revolves  to  a" ;  a"  b  is  equal  to  a'  b ;  the 
line  joining  a  a"  is  perpendicular  to  G  L 


PROJECTIONS.  85 

Remark.  The  preceding  figures  are  pictorial  representa- 
tions ;  in  the  remaining  figures  a  horizontal  line  is  used  to 
separate  the  planes  of  projection,  the  part  above  the  line  be- 
ing the  vertical  plane,  and  the  part  below  the  horizontal. 

It  is  important,  then,  to  note,  1st,  that  the  perpendicular 
distance  from  the  horizontal  projection  of  a  point  to  the 
ground  line  shows  how  far  the  point  itself  is  from  the  verti- 
cal plane  of  projection. 

2d,  that,  in  like  manner ',  the  perpendicular  distance  from 
its  vertical  projection  to  the  ground  line  shows  its  height 
above  the  horizontal  plane. 

3d,  that  the  horizontal  and  vertical  projections  of  a  point 
lie  on  the  right  line  drawn  from  one  to  the  other,  and  per- 
pendicular to  the  ground  line. 

4th,  that  the  distance,  measured  horizontally,  between  two 
points,  is  that  between  their  horizontal  projections. 

5th,  that  their  distance  apart  vertically,  or  the  height  the  one 
is  above  the  other,  is  measured  by  the  difference  between  the 
respective  distances  of  their  vertical  projections  from  the 
ground  line. 

6th,  that  the  actual  distance  between  two  points,  or  the  length 
of  a  right  line  connecting  them,  is  equal  to  the  hypothenuse 
of  a  right  angled  triangle,  the  base  of  which  is  equal  to  the 
distance  between  the  horizontal  projections  of  the  points,  and 
the  altitude  is  the  difference  between  the  distances  of  their 
vertical  projections  from  the  ground  line. 

For  example  (PI.  V.  Fig.  72),  the  line  G  L  is  the  ground 
line.  The  point  a  being  the  horizontal,  and  the  point  a'  the 
vertical  projection  of  a  point,  these  two  points  lie  on  the  right 
line  ax-a'  joining  them,  and  perpendicular  to  QL.  The  point 
itself  is  at  a  distance  in  front  of  the  vertical  plane  measured 
by  the  line  ax;  and  at  a  height  from  the  horizontal  plane 
measured  by  x  a'. 

In  like  manner  b  b',  and  c  cf,  are  the  projections  of  two 
points,  the  horizontal  distance  between  which  is  b  c,  the  dis- 
tance apart  of  their  horizontal  projections ;  and  the  vertical 
distance  is  c'y,  equal  to  the  difference  between  c'x'  and  b'x, 
their  respective  heights  from  the  horizontal  plane.  The  actual 
distance  between  these  points,  or  the  length  of  the  line  drawn 


86  INDUSTRIAL  DRAWING. 

from  one  to  the  other,  may  be  found  by  constructing  a  right 
angled  triangle,  mon /  the  base  of  which,  o  m,  being  equal  tc 
b  c,  and  its  altitude,  o  n,  equal  to  c'y,  its  hy pothenuse,  m  n,  will 
be  the  distance  required. 

In  making  the  projections  of  an  object,  when  it  is  desirable 
to  designate  the  projections  that  correspond  to  the  same  point, 
they  are  joined  by  a  light  broken  line  ;  and  if  the  projections 
are  those  of  an  isolated  point,  either  the  projections  are  made 
with  a  large  round  dot,  or  by  a  small  dot  surrounded  by  a 
small  circle.  When  the  projections  of  two  points  are  those  of 
the  extremities  of  a  right  line,  a  full  line  is  drawn  on  each 
plane  of  projection  between  the  points,  as  t>  c  and  Vd  ;  and  a 
broken  line  is  drawn  between  the  projections  of  the  corre- 
sponding extremities. 

Notation. 

Small  letters  are  used  to  designate  the  projections  of  a 
point,  the  same  letter  being  used  for  both  projections  ;  to  dis- 
tinguish between  them,  the  vertical  is  accented.  The  point  a  of 
is  also  spoken  of  as  the  point  A. 

Lines  are  similarly  treated,  as  the  line  oib — a'b',  or  the  line 
AB. 

The  letters  H  and  V  are  used  to  designate  the  planes  of 
projection. 

G  L  stands  for  the  ground  line. 

Shade  lines. 

Shade  lines  upon  outline  drawings  add  very  much  to  their 
appearance ;  when  properly  placed,  they  give  relief  to  the 
drawing,  and  are  of  assistance  in  reading  it. 

In  mechanical  drawing  the  light  is  generally  assumed  to 
come  in  such  a  direction  that  its  projections  shall  make  angles 
of  45°  with  the  ground  line ;  the  arrows  in  Figs.  23  and  24 
indicate  the  direction  of  light. 

Those  edges  should  be  heavy  which  separate  light  from 
dark  surfaces. 

In  the  case  of  the  cube  (Fig.  23),  we  see  that  the  top,  front, 
and  left-hand  faces  must  be  in  the  light,  while  the  remaining 
faces  would  be  in  the  shade.  In  the  elevation  the  only  visible 


PEOJECTION8. 


87 


edges  separating  light  from  dark  faces  are  those  upon  the 
right  and  lower  sides  (a'V  and  5V),  while  in  the  plan  the 
shaded  edges  are  upon  the  right  and  upper  sides  (0/and/e). 


Fi».  23.  FKJ.  24. 

In  case  of  a  curved  surface  like  the  cylinder  (Fig.  24),  the 
line  a'b'  does  not  separate  the  light  from  the  dark  surface, 
yet  it  is  well  to  make  it  a  trifle  darker  than  the  left-hand  edge, 
but  not  as  dark  as  the  bottom  line  c'V.  In  the  plan  the  circle 
is  made  darkest  upon  the  upper  and  right-hand  side,  tapering 
to  the  points  of  tangency  (e,tf)  of  rays  of  light. 

Shade  lines  of  sections  follow  the  above  rules,  as  shown  in 
Fig.  83.  PI.  YI. 

Draw  the  shade  lines  with  their  breadth  outside  the  outline. 

In  colored  drawings  draw  the  shade  lines  last. 

In  shaded  drawings  omit  the  shade  lines  altogether. 

Profiles  and  Sections.  The  projections  of  an  object  give 
only  the  forms  and  dimensions  of  its  exterior,  and  the  posi- 
tions of  points,  &c.,  on  its  surface.  To  show  the  thickness  of 
its  solid  parts,  and  the  form  and  dimensions  of  its  interior, 
intersecting-planes  are  used.  Taking  a  house  as  a  model, 
let  us  conceive  it  to  be  cut,  or  sawed  through,  at  some 
point  between  its  two  ends,  in  the  direction  of  a  vertical 
plane  parallel  to  the  ends.  Setting  aside  one  portion,  let 
us  imagine  a  pane  of  glass  placed  against  the  sawed  surface 
of  the  other,  and  let  an  accurate  outline  of  the  parts  thus 
cut  through  be  traced  on  the  pane.  This  outline  is  termed 


88  INDUSTRIAL   DRAWING. 

a  profile.  On  it,  to  distinguish  the  solid  parts  cut  through 
from  the  voids,  or  hollow  parts,  we  cover  them  entirely 
with  ink,  or  some  other  color,  or  else  simply  draw  par- 
allel lines  close  together  across  them.  If,  besides  tracing 
the  outline  of  the  parts  resting  against  the  pane,  we  were  to 
trace  the  projections  of  all  the  parts,  both  within  and  without 
the  outline  of  the  profile,  that  could  be  seen  through  the 
pane  by  a  person  standing  in  front  of  it,  the  profile  with 
these  additional  outlines  is  termed  a  section.  A  section  more- 
over differs  from  a  profile  in  this,  that  it  may  be  made  in  any 
direction,  whereas  the  profile  is  made  by  cutting  vertically, 
and  in  objects,  like  a  house,  bounded  by  plane  surfaces,  in  a 
direction  perpendicular  to  the  surface. 

To  show  the  direction  in  which  the  section  is  made,  it  is 
usual  to  draw  a  broken  and  dotted  line  on  the  plan  and  eleva- 
tion of  the  object,  marking  the  position  of  the  saw-cut  on  the 
surface  of  the  object ;  and,  to  indicate  the  position  to  which 
the  section  corresponds,  letters  of  reference  are  placed  at  the 
extremities  of  each  of  these  lines,  and  the  figure  of  the  section 
is  designated  as  vertical,  or  oblique  section  on  A — J2,  C — D, 
&c.,  according  as  the  section  is  in  a  plane  perpendicular,  or 
oblique  to  the  horizontal  plane  of  projection. 

The  sections  in  most  general  use  are  those  made  by  vertical 
and  horizontal  planes.  A  horizontal  section  is  made  in  the 
same  manner  as  a  vertical  one,  by  conceiving  the  object  cut 
through  at  some  point  above  the  horizontal  plane  of  projec- 
tion, and  parallel  to  it,  and,  having  removed  the  portion  above 
the  plane  of  section,  by  making  such  a  representation  of  the 
lower  portion  as  would  be  represented  by  tracing  on  a  pane 
of  glass,  laid  on  it,  the  outline  of  the  parts  in  contact  with 
the  pane,  with  the  outline  of  the  projection  of  the  parts  on  the 
pane  that  can  be  seen  through  it,  whether  on  the  exterior,  or 
interior  of  the  object. 

The  solid  parts  in  contact  with  the  pane  are  represented  in 
the  same  way  as  in  other  sections.  The  projected  parts  are 
represented  only  by  their  outlines.  A  broken  and  dotted  line, 
with  letters  of  reference  at  its  extremities,  is  drawn  on  the 
elevation  to  show  where  the  section  is  taken ;  and  the  section 
is  designated  by  a  title,  as,  horizontal  section  on  A — £,  &c. 


PROJECTIONS.  89 

As  the  broken  and  dotted  lines  that  indicate  the  position 
of  the  planes  of  section  are  drawn  on  the  planes  of  projection, 
and  are  in  fact  the  lines  in  which  the  planes  of  section  would 
cut  these  two  planes,  they  are  termed  the  vertical  or  hori- 
zontal traces  of  the  planes  of  section,  according  as  the  lines 
are  traced  on  the  vertical  or  horizontal  plane  of  projection. 

It  will  be  well  to  note  particularly  that  the  planes  of  section 
are  usually  taken  in  front  of,  or  above  the  object,  that  portion 
of  it  which  is  cut  by  the  plane  being  supposed  in  contact  with 
the  plane ;  whereas  the  planes  of  projection  may  be  placed 
either  behind,  or  in  front  of  the  object,  and  above  or  below  it, 
as  may  best  suit  the  purpose  of  the  draftsman ;  the  position 
of  the  ground  line  therefore  will  always  indicate  on  which 
side  of  the  object,  and  whether  above,  or  below  it,  the  planes 
of  projection  are  placed.  The  usual  method  is  to  place  the 
horizontal  plane  of  projection  below  the  object  represented 
and  the  vertical  plane  behind  it.  The  more  usual  method 
also  is  to  represent  the  object  as  resting  on  the  horizontal 
plane  ;  its  position  with  respect  to  the  vertical  plane,  or  thai 
of  the  vertical  plane  with  respect  to  it,  being  so  taken  as  to 
give  the  desired  elevation  to  suit  the  views  of  the  draftsman. 
In  the  case  of  the  ordinary  house,  for  example,  the  elevations 
of  the  four  sides  may  be  obtained  either  by  supposing  one 
vertical  plane,  and  the  four  sides  successively  presented  to  it ; 
or  by  supposing  the  vertical  plane  shifted  so  as  to  be  brought 
behind  each  of  the  sides  in  succession. 

Projections  of  Points  and  Right  Lines.  The  method  of 
projections  presents  two  problems.  The  one  is  having  given 
the  forms  and  dimensions  of  an  object,  to  construct  its  pro- 
jections ;  the  other,  having  the  projections  of  an  object,  to 
construct  its  forms  and  dimensions.  A  correct  understanding 
of  the  manner  of  projecting  points  and  right  lines,  and 
determining  their  relative  positions  with  respect  to  each 
other,  is  an  indispensable  foundation  for  the  solution  of  these 
two  questions. 

The  methods  of  projecting  a  single  point,  and  of  obtaining 
its  distance  from  the  planes  of  projection,  also  of  two  pointe; 
and  determining  their  distance  apart,  have  already  been 
given.  The  same  process  would  evidently  be  followed  in 


90  INDUSTRIAL   DRAWING. 

projecting  any  number  of  points ;  or,  in  determining  their 
relative  positions,  having  their  projections.  But,  besides 
these  general  methods,  there  are  some  particular  cases  with 
which  it  will  be  well  to  become  familiarized  at  the  outset,  as 
a  knowledge  of  them  will  materially  aid  in  showing,  by  a 
glance  at  the  projections,  the  relative  positions  of  the  lines 
joining  the  points  to  the  planes  of  projection ;  that  is,  whether 
these  fines  are  parallel,  oblique,  or  perpendicular  to  one,  or 
both  of  these  planes. 

Case  1.  (PI.  V.  Fig.  73.)  Let  aa'  and  bV  be  the  projec- 
tions of  two  points,  the  distances  of  their  vertical  projections 
a'x  and  b'x'  from  the  ground  line  being  equal,  those  of  their 
horizontal  projections  ax  and  bx'  being  unequal.  The  points 
themselves  will  be  at  the  same  height  above  the  horizontal 
plane  of  projection  but  at  unequal  distances  from  the  vertical 
plane.  The  vertical  projection  of  the  line  joining  the  two 
points  a'b'  will  be  parallel  to  the  ground  line,  and  its  hori- 
zontal projection  ab  will  be  oblique  to  it. 

from  this  we  observe,  that  when  two  points  of  a  right  line 
are  at  the  same  height  above  tJie  horizontal  plane  of  projection, 
and  at  unequal  distances  from  the  vertical  plane,  the  vertical 
projection  of  the  line  will  be  parallel  to  the  ground  line,  and 
its  horizontal  projection  oblique  to  this  line. 

Finding  then  the  two  projections  of  a  right  line  in  these 
positions  with  respect  to  the  ground  line,  we  conclude  that 
the  line  itself  is  at  the  same  height  throughout  above  the 
horizontal  plane  of  projection,  or  parallel  to  this  plane,  but 
oblique  to  the  vertical  plane. 

Case  2.  (PI.  Y.  Fig.  74.)  In  like  manner,  when  we  find 
the  horizontal  projections  of  two  points  a  and  b  at  the  same 
distance  from  the  ground  line,  and  the  vertical  projections 
a'  and  b'  at  unequal  heights  from  it,  we  conclude  that  the  line 
joining  the  points  is  parallel  to  the  vertical  plane  but  oblique 
to  the  horizontal. 

Case  3.  (PI.  V.  Fig.  75.)  When  the  horizontal  projections 
of  two  points  are  at  the  same  distance  from  ground  line,  and  the 
vertical  projections  also  at  equal  distances  from  it,  we  conclude 
that  the  line  itself  is  parallel  to  both  planes  of  projection. 

When  a  line  therefore  is  parallel  to  one  plane  of  projection 


PBOJECTION8.  91 

(done,  its  projection  on  the  other  willbe  parallel  to  the  ground 
line,  and  its  projection  on  the  plane  to  which  it  is  parallel 
will  be  oblique  to  the  ground  line. 

When  the  line  is  parallel  to  loth  planes  its  two  projections 
wiU  be  parallel  to  the  ground  line. 

Case  4.  (PL  Y.  Fig.  76.)  Suppose  two  points  as  a  and  b 
to  lie  in  the  horizontal  plane  of  projection,  where  are  their 
vertical  projections?  From  what  has  been  already  shown, 
these  last  projections  must  lie  on  the  perpendiculars,  from  the 
horizontal  projections  a  and  b  to  the  ground  line ;  but  as  the 
points  are  in  the  horizontal  plane  their  projections  cannot  lie 
above  the  ground  line.  The  vertical  projections  of  a  and  b 
therefore  must  be  at  a'  and  b'  on  the  ground  line,  where  the 
perpendiculars  from  a  and  b  cut  it.  For  a  like  reason  the 
vertical  projection  of  a  line  as  a — b  in  the  horizontal  plane 
will  be  as  a' — b'  in  the  ground  line. 

In  like  manner  the  horizontal  projections  of  points  and 
lines  lying  in  the  vertical  plane  of  projection  will  be  also  in 
the  ground  line. 

Case  5.  (PL  Y.  Fig.  77.)  If  a  line  is  vertical,  orperpen 
dicular  to  the  horizontal  plane  of  projection,  its  projection 
on  that  plane  will  be  a  point  simply,  as  a.  For,  the  line  be- 
ing vertical,  if  a  plumb  line  were  applied  along  it  the  two 
lines  would  coincide,  and  the  point  of  the  bob  of  the  plumb 
line  would  indicate  only  one  point  as  the  projection  of  the 
entire  line.  Now  as  a  is  the  horizontal  projection  of  all  the 
points  of  the  line,  their  vertical  projections  must  lie  in  the 
line  from  a  perpendicular  to  the  ground  line,  so  that  the  ver- 
tical projections  of  any  two  points  of  the  vertical  line  at  the 
given  heights  b'x,  and  a'x  above  the  horizontal  plane  of  pro- 
jection, would  be  projected  on  the  perpendicular  from  a  to 
the  ground  line,  and  at  the  given  distances  b'x  and  a'x  above 
the  ground  line. 

In  like  manner  it  can  be  shown,  that  a  line  perpendicular 
to  the  vertical  plane  is  projected  into  a  point,  as  a',  and  its 
horizontal  projection  will  lie  on  the  perpendicular  to  the 
ground  line  from  a',  as  a—  b,  in  which  the  distances  of  the 
points  a  and  b  from  the  ground  line  show  the  distances  of  the 
ends  of  the  line  from  the  vertical  plane. 


90  INDUSTRIAL   DRAWING. 

projecting  any  number  of  points ;  or,  in  determining  their 
relative  positions,  having  their  projections.  But,  besides 
these  general  methods,  there  are  some  particular  cases  with 
which  it  will  be  well  to  become  familiarized  at  the  outset,  as 
a  knowledge  of  them  will  materially  aid  in  showing,  by  a 
glance  at  the  projections,  the  relative  positions  of  the  lines 
joining  the  points  to  the  planes  of  projection ;  that  is,  whether 
these  lines  are  parallel,  oblique,  or  perpendicular  to  one,  or 
both  of  these  planes. 

Case  1.  (PI.  V.  Fig.  73.)  Let  aa!  and  bb'  be  the  projec- 
tions of  two  points,  the  distances  of  their  vertical  projections 
a'x  and  b'x'  from  the  ground  line  being  equal,  those  of  their 
horizontal  projections  ax  and  bx'  being  unequal.  The  points 
themselves  will  be  at  the  same  height  above  the  horizontal 
plane  of  projection  but  at  unequal  distances  from  the  vertical 
plane.  The  vertical  projection  of  the  line  joining  the  two 
points  a'b'  will  be  parallel  to  the  ground  line,  and  its  hori- 
zontal projection  db  will  be  oblique  to  it. 

from  this  we  observe,  that  when  two  points  of  a  right  line 
are  at  the  same  height  above  the  horizontal  plane  of 'projection, 
and  at  unequal  distances  from  the  vertical  plane,  the  vertical 
projection  of  the  line  will  be  parallel  to  the  ground  line,  and 
its  horizontal  projection  oblique  to  this  line. 

Finding  then  the  two  projections  of  a  right  line  in  these 
positions  with  respect  to  the  ground  line,  we  conclude  that 
the  line  itself  is  at  the  same  height  throughout  above  the 
horizontal  plane  of  projection,  or  parallel  to  this  plane,  but 
oblique  to  the  vertical  plane. 

Case  2.  (PI.  Y.  Fig.  74.)  In  like  manner,  when  we  find 
the  horizontal  projections  of  two  points  a  and  b  at  the  same 
distance  from  the  ground  line,  and  the  vertical  projections 
a'  and  b'  at  unequal  heights  from  it,  we  conclude  that  the  line 
joining  the  points  is  parallel  to  the  vertical  plane  but  oblique 
to  the  horizontal. 

Case  3.  (PI.  V.  Fig.  75.)  When  the  horizontal  projections 
of  two  points  are  at  the  same  distance  from  ground  line,  and  the 
vertical  projections  also  at  equal  distances  from  it,  we  conclude 
that  the  line  itself  is  parallel  to  both  planes  of  projection. 

When  a  line  therefore  is  parallel  to  one  plane  of  projection 


PROJECTIONS.  91 

alone,  its  projection  on  the  other  willle  parallel  to  the  ground 
line,  and  its  projection  on  the  plane  to  which  it  is  parallel 
will  be  oblique  to  the  ground  line. 

When  the  line  is  parallel  to  both  planes  its  two  projections 
Witt  be  parallel  to  the  ground  line. 

Case  4.  (PI.  V.  Fig.  76.)  Suppose  two  points  as  a  and  b 
to  lie  in  the  horizontal  plane  of  projection,  where  are  their 
vertical  projections?  From  what  has  been  already  shown, 
these  last  projections  must  lie  on  the  perpendiculars,  from  the 
horizontal  projections  a  and  b  to  the  ground  line ;  but  as  the 
points  are  in  the  horizontal  plane  their  projections  cannot  lie 
above  the  ground  line.  The  vertical  projections  of  a  and  b 
therefore  must  be  at  a'  and  b'  on  the  ground  line,  where  the 
perpendiculars  from  a  and  b  cut  it.  for  a  like  reason  the 
vertical  projection  of  a  line  as  a — b  in  the  horizontal  plane 
will  be  as  a' — b'  in  the  ground  line. 

In  like  manner  the  horizontal  projections  of  points  and 
lines  lying  in  the  vertical  plane  of  projection  will  be  also  in 
the  ground  Line. 

Case  5.  (PL  V.  Fig.  77.)  If  a  line  is  vertical,  or  perpen 
dicular  to  the  horizontal  plane  of  projection,  its  projection 
on  that  plane  will  be  a  point  simply,  as  a.  For,  the  line  be- 
ing vertical,  if  a  plumb  line  were  applied  along  it  the  two 
lines  would  coincide,  and  the  point  of  the  bob  of  the  plumb 
line  would  indicate  only  one  point  as  the  projection  of  the 
entire  line.  Now  as  a  is  the  horizontal  projection  of  all  the 
points  of  the  line,  their  vertical  projections  must  lie  in  the 
line  from  a  perpendicular  to  the  ground  line,  so  that  the  ver- 
tical projections  of  any  two  points  of  the  vertical  line  at  the 
given  heights  b'x,  and  a'x  above  the  horizontal  plane  of  pro- 
jection, would  be  projected  on  the  perpendicular  from  a  to 
the  ground  line,  and  at  the  given  distances  b'x  and  a'x  above 
the  ground  line. 

In  like  manner  it  can  be  shown,  that  a  line  perpendicular 
to  the  vertical  plane  is  projected  into  a  point,  as  a',  and  its 
horizontal  projection  will  lie  on  the  perpendicular  to  the 
ground  line  from  a',  as  a — b,  in  which  the  distances  of  the 
points  a  and  b  from  the  ground  line  show  the  distances  of  the 
ends  of  the  line  from  the  vertical  plane. 


94  INDUSTEIAL  DRAWING. 

3d.  Draw  lines  from  o'  to  the  points  a'Vc'd'. 

Remarks.  The  line  o'd'  in  vertical  projection,  and  the 
line  OG,  according  to  what  has  been  laid  down,  should  be 
dotted. 

Having  the  projections  of  a  like  pyramid  we  would  pro- 
ceed, as  in  the  last  case,  to  construct  its  edges  and  faces  if  re- 
quired for  a  model. 

Prob.  74.  (PI.  VI.  Fig.  81.)  To  construct  the  projections 
of  a  right  prism  with  a  regular  hexagonal  base. 

Let  the  base  of  the  prism  be  supposed  to  rest  on  the  hori- 
zontal plane  of  projection. 

1st.  Construct  at  a  convenient  distance  from  the  ground 
line  the  regular  hexagon  abc,  &c.,  of  the  base ;  taking  two  of 
its  opposite  sides,  as  o — c,  &nd.f— e,  parallel  to  this  line. 

2d.  Construct  the  projections  h'l'm',  &c.,  of  the  points 
abc,  &c. 

3d.  As  the  edges  of  the  prism  are  vertical,  their  vertical 
projections  will  be  drawn  through  the  points  AT,  &c.,  and 
perpendicular  to  the  ground  line. 

4th.  Having  drawn  these  lines,  set  off  the  equal  distances 
h' — a',  I' — 5',  &c.,  upon  them,  and  each  equal  to  the  height 
of  the  prism. 

Remarks.  As  the  edges  projected  in  b  and  c,  and  f  and  <?, 
are  projected  in  the  vertical  lines  l'—bf  and  m' — c'  the  back 
edges  cannot  be  represented  by  dotted  lines. 

As  the  top  of  the  prism  is  parallel  to  its  base  it  is  projected 
vertically  in  the  line  a' — d'  equal  and  parallel  to  h' — n'}  the 
projection  of  the  base. 

All  the  faces  of  the  prism  are  equal  rectangles,  and  each 
equal  to  Vdm'l'  the  projection  of  the  face  parallel  to  the  ver- 
tical plane. 

It  has  been  shown,  in  the  projections  of  right  lines,  that 
when  a  line  is  parallel  to  one  plane  of  projection,  its  projec- 
tion on  that  plane  is  equal  to  the  length  of  the  line,  and  that 
its  projection  on  the  other  plane  is  parallel  to  the  ground 
line.  In  this  case  we  see  that  the  top  of  the  prism,  which  is 
a  hexagon  parallel  to  the  horizontal  plane,  is  projected  on 
that  plane  into  the  equal  hexagon  abc,  &c.,  and  on  the  verti- 
cal plane  into  a  line  a' — d'  parallel  to  the  ground  line.  In 


PBOJECTIONS.  95 

like  manner  we  see  that  the  face  of  which  ~b — G  is  the  horizon- 
tal projection  and  b'c'm'l'  the  vertical,  is  projected  on  the 
horizontal  plane  in  a  line  parallel  to  the  ground  line,  and  on 
the  vertical  plane  in  a  rectangle  equal  to  itself.  From  this 
we  conclude >,  that  when  a  plane  figure  is  parallel  to  one  plane 
of  projection  it  will  be  projected  on  that  plane  in  a  figure 
equal  to  itself ,  and  on  the  other  plane  into  a  line  parallel  to 
the  ground  line.  Moreover  since  the  faces  of  the  prism  are 
plane  surfaces  perpendicular  to  the  horizontal  plane,  and  are 
projected  respectively  into  the  lines  a — b,  b — c,  &c.,  we  con- 
clude that  a  plane  surf  ace  perpendicular  to  one  plane  of  pro- 
jection is  projected  on  that  plane  into  a  right  line.  The 
same  is  true  of  the  base  and  top  of  the  prism ;  the  base  being 
in  the  horizontal  plane,  which  is  perpendicular  to  the  vertical 
plane,  is  projected  into  the  ground  line  in  h' — n' ;  the  top 
being  parallel  to  the  horizontal  plane  is  likewise  perpendicu- 
lar to  the  vertical  plane,  and  is  projected  into  the  line 
a'—d'. 

Traces  of  Planes  on  the  Planes  of  Projection.  The  plane 
surfaces  of  the  prism  and  pyramids  in  the  preceding  problems 
being  of  limited  extent,  we  have  only  had  to  consider  the 
lines  in  which  they  cut  the  horizontal  plane  of  projection,  as 
a — b,  b — c,  &c.,  the  bounding  lines  of  the  bases  of  these  solids 
These  lines  are  therefore  properly  the  traces  of  these  limited 
plane  surfaces  on  the  horizontal  plane.  But  when  a  plane  is 
of  indefinite  extent,  we  may  have  to  consider  the  lines  in 
which  it  cuts,  or  meets  both  planes  of  projection.  The  most 
usual  cases  in  which  we  have  to  consider  these  lines  are  in 
those  of  profile  planes,  and  planes  of  section,  in  which  the 
planes  are  perpendicular  either  to  the  horizontal,  or  vertical 
plane,  and  parallel,  or  oblique  to  the  other. 

The  position  of  the  trace  of  a  plane,  when  parallel  to  one 
plane  of  projection  and  perpendicular  to  the  other,  as  has 
already  been  shown,  is  a  line  parallel  to  the  ground  line,  and 
on  that  plane  of  projection  to  which  the  plane  is  perpendicu- 
lar, as  the  lines  b — c,  and  f- — e,  for  example,  which  are  the 
traces  on  the  horizontal  plane  of  the  faces  of  the  prism,  which 
are  perpendicular  to  this  plane,  and  parallel  to  the  vertical 
plane.  The  same  may  be  said  of  the  line  a'— d'y  which  would 


96  INDUSTRIAL   DRAWING. 

be  the  trace  of  the  plane  of  the  top  of  the  prism,  if  it  -were 
produced  back  to  meet  the  vertical  plane. 

When  the  plane  is  perpendicular  to  the  horizontal  plane^ 
but  oblique  to  the  vertical,  as  for  example  the  face  of  the 
prism  of  which  a — -f,  or  d — e  is  the  horizontal  trace,  its  verti- 
cal trace  will  be  perpendicular  to  the  ground  line  at  the  point 
where  the  horizontal  trace  meets  this  line.  To  show  this,  sup- 
pose the  prism  so  placed  as  to  have  its  back  face  against  the 
vertical  plane ;  then  the  line  a— -f,  for  example,  will  be  oblique 
to  the  ground  line,  the  point  f  of  this  line  being  on  it,  at 
the  point  I',  whilst  the  line  I' — b',  the  one  in  which  the  oblique 
face  meets  the  vertical  plane,  or  its  trace  on  this  plane,  will 
be  perpendicular  to  the  ground  line.  The  same  illustration 
would  hold  true  supposing  the  prism  laid  on  one  of  its  faces 
on  the  horizontal  plane,  with  its  base  against,  the  vertical  plane. 

If  A— B  (PL  VI.  Fig.  82)  therefore  represents  the  horizon- 
tal trace  of  a  plane  perpendicular  to  the  horizontal  plane  its 
vertical  trace  Witt  be  a  line  B — b,  drawn  from  the  point  B> 
where  the  horizontal  trace  cuts  the  ground  line,  perpendicu- 
lar to  this  line.  In  like  manner,  if  E—F is  the  vertical 
trace  of  a  plane  perpendicular  to  the  vertical  plane  and 
oblique  to  the  horizontal  plane,  the  line  F—f  perpendicular 
to  the  ground  line  is  its  horizontal  trace. 

Prob.  75.  (PI.  VI.  Fig.  83.)  To  construct  the  projections 
and  sections  of  a  hollow  cube  of  given  dimensions. 

Let  us  suppose  the  cube  so  placed  that,  its  base  resting  on 
the  horizontal  plane  of  projection  in  front  of  the  vertical 
plane,  its  front  and  back  faces  shall  be  parallel  to  the  vertical 
plane,  and  its  other  two  ends  perpendicular  to  this  plane. 

Having  constructed  a  square  abed  (Fig.  X)  of  the  same 
dimensions  as  the  base  of  the  given  cube,  and  having  its  sides 
ab  and  cd  parallel  to  the  ground  line,  and  at  any  convenient 
distance  from  it;  this  square  may  be  taken  as  the  projection 
of  the  base  of  the  cube.  But  as  the  top  of  the  cube  is  paral- 
lel to  the  base,  and  its  four  faces  are  also  perpendicular  to 
these  two  parts,  the  top  will  be  also  projected  into  the  square 
abed,  and  the  four  sides  respectively  into  the  sides  of  the 
square.  The  square  abed  will  therefore  be  the  horizontal  pro- 
jection of  all  the  exterior  faces  of  the  cube. 


PROJECTIONS.  97 

Having  projected  the  base  of  the  cube  into  the  vertical 
plane,  which  projection  (Fig.  Y)  will  be  a  line  h' — Z',  on  the 
ground  line,  equal  to  a— b,  construct  the  square  a'Vlh'  equal 
to  abed.  This  is  the  vertical  projection  of  the  cube. 

As  the  interior  faces  of  the  cube  cannot  be  seen  from  with- 
out, the  following  method  is  adopted  to  represent  their  pro- 
jections :  within  the  square  abed  construct  another  represented 
by  the  dotted  lines,  having  its  sides  at  the  same  distance  from 
the  exterior  square  as  the  thickness  of  the  sides  of  the  hollow 
cube.  This  square  will  be  the  projection  of  the  interior  faces ; 
and  it  is  drawn  with  dotted  lines,  to  show  that  these  faces  are 
not  seen  from  without. 

Supposing  the  top  and  bottom  of  the  cube  of  the  same 
thickness  as  the  sides,  a  like  square  constructed  within  a'b'l'h' 
will  be  the  vertical  projection  of  the  two  interior  faces,  which 
are  perpendicular  to  the  vertical  plane,  and  of  the  interior 
faces  of  the  top  and  base. 

Having  completed  the  projections  of  the  cube,  suppose  it 
is  required  to  construct  the  figures  of  the  sections  cut  from  it 
by  a  horizontal  plane,  of  which  M — N  is  the  vertical  trace ; 
and  by  a  vertical  plane  of  which  O^-P  is  the  horizontal 
trace.  The  horizontal  plane  of  section  will  cut  from  the 
exterior  faces  of  the  cube  a  square  mopn  (Fig.  Z]  equal  to  the 
one  abed,  and  from  the  interior  faces  another  square  equal  to 
the  one  in  dotted  lines,  and  having  its  sides  parallel  to  those 
of  mopn.  The  solid  portion  of  the  four  sides  cut  by  the 
plane  of  section  would  be  represented  by  the  shading  lines, 
as  in  Fig.  Z. 

The  plane  of  section  of  which  O — P  is  the  trace,  being 
oblique  to  the  sides,  will  cut  from  the  opposite  exterior  faces 
a — d,  and  b — c,  and  the  exterior  faces  of  the  top  and  base,  a 
rectangle  of  which  r — u  (Figs.  X,  W)  is  the  base,  and  r — / 
(Fig.  TF)  equal  to  the  height  V — Z',  is  the  altitude ;  in  like 
manner  it  will  cut  from  the  corresponding  interior  faces  a 
rectangle  of  which  s — t  is  the  base,  and  s—s',  equal  to  the 
height  of  the  interior  face,  is  the  altitude.  The  sides  of  the 
interior  rectangle  stt's'  will  be  parallel  to  those  of  the  one 
exterior ;  the  distance  apart  of  the  vertical  sides  being  equal 
to  the  equal  distances  r—s  (Fig.  X]  and  t— u  ;  and  that  of 
1 


98  INDUSTRIAL   DRAWING. 

the  horizontal  sides  being  the  same  as  the  thickness  of  the 
top  and  base  of  the  cube.  In  other  words,  as  has  already 
been  stated,  the  figure  of  the  vertical  section  is  the  same  that 
would  be  found  by  tracing  the  outline  of  the  part  of  the 
cube,  cut  through  by  the  plane  of  section,  on  the  vertical 
plane  of  projection. 

Remarks.  The  manner  of  representing  the  interior  faces 
of  the  hollow  cube  by  dotted  lines  is  generally  adopted  for 
all  like  cases ;  that  is,  when  it  is  desired  to  represent  the  pro- 
jections of  the  outlines  of  any  part  of  an  object  which  lies 
between  some  other  part  projected  and  the  plane  of  pro- 
jection. 

Where  several  points,  situated  on  the  same  right  line,  as 
r,  «,  t,  u,  on  the  line  O — P  (Fig.  JT),  are  to  be  transferred  to 
another  right  line,  as  in  the  construction  of  (Fig.  TF),  the 
shortest  way  of  doing  it,  and,  if  care  be  taken,  also  the  most 
accurate  one,  is  to  place  the  straight  edge  of  a  narrow  strip 
of  paper  along  the  line,  and  confining  it  in  this  position  to 
mark  accurately  on  it  near  the  edge  the  positions  of  the 
points.  Having  done  this,  the  points  can  be  transferred  from 
the  strip,  by  a  like  process,  to  any  other  line.  The  advantage 
of  this  method  over  that  of  transferring  each  distance  by  the 
dividers  will  be  apparent  in  some  of  the  succeeding  problems. 

Prob.  76.  (PL  VI.  Fig.  84.)  To  construct  the  projec- 
tions of  a  regular  hollow  pyramid  truncated  by  a  plane 
oblique  to  the  horizontal  plane  and  perpendicular  to  the  ver- 
tical plane. 

Let  us  suppose  the  base  of  the  pyramid  a  regular  pentagon. 
Having  constructed  this  base,  and  the  projections  of  the 
different  parts  of  the  entire  pyramid  as  in  Prob.  71,  draw  a 
line,  M—N,  oblique  to  the  ground  line,  as  the  vertical  trace 
of  the  assumed  truncating  plane  ;  the  portion  of  the  pyramid 
lying  above  this  plane  being  supposed  removed. 

Now,  as  the  truncating  plane  cuts  all  the  faces  of  the 
pyramid,  and  as  it  is  itself  perpendicular  to  the  vertical  plane 
of  projection,  all  the  lines  which  it  cuts  from  these  faces  will 
be  projected  on  the  vertical  plane  of  projection  in  the  trace 
M- — N.  The  points  r',  s',  t ',  v',  and  u',  where  the  trace 
M — -STcuts  the  projections  of  the  edges  of  the  pyramid,  will 


PROJECTIONS.  99 

be  the  projections  of  the  points  in  which  the  truncating  plane 
cuts  these  edges ;  and  the  line  rr — v'  for  example  is  the  projec- 
tion of  the  line  cut  from  the  exterior  face  projected  in  ~b'v'ti '. 

The  horizontal  projections  of  the  points  of  which  r',  v\ 
&c.,  are  the  vertical  projections,  will  be  found  on  the  hori- 
zontal projections  vc,  vb,  &c.,  of  the  edges  of  which  v'c',  v'b', 
&c.,  are  the  vertical  projections,  and  will  be  obtained  in  the 
usual  way.  Joining  the  corresponding  points  r,  v,  s,  &c., 
thus  obtained,  the  figure  rstuv,  will  be  the  horizontal  projec- 
tion of  the  one  cut  from  the  exterior  faces  of  the  pyramid  by 
the  truncating  plane. 

Thus  far  nothing  has  been  said  of  the  projections  and 
sections  of  the  interior  faces  of  the  pyramid.  To  construct 
these  let  us  take  the  thickness  of  the  sides  of  the  hollow 
pyramid  to  be  the  same,  in  which  case  the  interior  faces  will 
be  parallel  to  and  all  at  the  same  distance  from  the  exterior 
faces.  If  another  pyramid  therefore  were  so  formed  as  to  fit 
exactly  the  hollow  space  within  the  given  one,  its  faces  and 
edges  would  be  parallel  to  the  corresponding  exterior  faces 
and  edges  of  the  given  hollow  pyramid,  and  its  vertex  would 
likewise  be  on  the  perpendicular  from  the  vertex  of  the  given 
pyramid  to  its  base.  Constructing,  therefore,  a  pentagon, 
mnopq,  having  its  sides  parallel  to,  and  at  the  same  distance 
from  those  of  dbcde,  this  figure  may  be  assumed  as  the  base 
of  the  interior  pyramid.  The  horizontal  projections  of  its 
edges  will  be  the  lines  vm,  vn,  &c.  To  find  the  vertical  pro- 
jections of  these  lines,  which  will  be  parallel  to  the  vertical 
projections  of  the  corresponding  exterior  edges,  project  the 
points  TO,  n,  &c.,  into  the  ground  line,  at  ra',  n',  &c.,  and, 
from  these  last  points,  draw  the  lines  m'v",  n'v",  parallel  to 
the  corresponding  ones  a'v',  b'v',  &c. ;  the  lines  m'v",  &c., 
will  be  the  required  projections. 

To  obtain  the  horizontal  projection  of  the  figure  cut  from 
the  interior  faces  by  the  plane  of  section,  find  the  points 
s,  y,  x,  &c.,  in  horizontal  projection,  corresponding  to  the 
points  zr,  y',  &c.,  in  vertical  projection,  where  the  trace 
M—N  cuts  the  lines  m'v",  n'v",  &c. ;  joining  these  points 
the  pentagon  zyx,  &c.,  will  be  the  required  horizontal  projec- 
tion. 


100  INDUSTEIAL   DRAWING. 

Having  constructed  the  projections  of  the  portion  of  the 
pyramid  below  the  truncating  plane,  Jet  it  now  be  required 
to  obtain  a  section  of  this  portion  by  a  vertical  plane  of  sec- 
tion through  the  vertex.  For  this  purpose,  to  avoid  the  con- 
fusion of  a  number  of  lines  on  the  same  drawing,  let  us 
construct  (Fig.  85)  another  figure  of  the  projections  of  the 
outlines  of  the  faces,  &c.  Having  drawn  the  line  O — P  for 
the  trace  of  the  vertical  section  through  the  projection  of  the 
vertes,  we  observe  that  this  plane  cuts  the  base  of  the  hollow 
pyramid  en  the  left-hand  side,  in  the  line  a — 3,  and  on  the 
opposite  slie  in  the  line  in — n;  setting  off  the  distances 
(Fig.  86)  a' — b'j  V — n',  and  n' — mf  on  the  ground  line, 
respectively  equal  to  a — J,  &c.,  we  obtain  the  line  of  section 
cut  from  the  ba&e.  Now  the  plane  of  section  cuts  the  exterior 
line  of  the  top  (Fig.  85)  on  the  left-hand  side,  in  a  point  hori- 
zontally projected  in  c,  and  vertically  in  c",  the  height  of 
which  point  above  the  base  is  the  distance  c' — c" ;  in  like 
manner  the  plane  cuts  the  interior  line  of  the  top,  on  the  same 
aide,  in  the  point  projected  horizontally  in  d,  and  vertically 
in  d",  its  height  above  the  base  being  d' — d".  The  corre- 
gponding  points  of  the  top  on  the  opposite  side  are  those  pro- 
lected  in  0,  o" ;  and  p,  p"  ;  their  corresponding  distances 
above  the  base  being  respectively  o' — o"  and  p'—^p". 

Having  thus  found  the  horizontal  and  vertical  distances 
between  these  points,  it  is  easy  to  construct  their  positions  in 
the  plane  of  section.  To  do  this,  set  off  on  the  ground  line 
(Fig.  86)  the  distances  a' — c',  a — d',  m' — o',  and  m! — -p'y 
respectively  equal  to  the  equal  corresponding  distances  a — c, 
&c.  (Fig.  85),  on  O — P.  At  the  povnts  c',  d',p',  and  o'  draw 
perpendiculars  to  the  ground  lino,  on  which  set  off  the  dis- 
tances c' — c",  &c.,  respectively  equal  to  those  c' — c"  of  Fig. 
85.  Having  drawn  tLe  lines  a' — c",  t" — d"y  and  V — d",  the 
figure  a'Vd"c"  is  the  section  of  the  left-hand  side ;  in  like 
manner  m!n'p"o"  is  the  figure  cut  fn.m  the  opposite  face  by 
the  plane  of  section. 

As  the  (Fig.  86)  represents  a  section,  and  not  a  profile  of 
the  pyramid,  we  must  draw  upon  it  the  lines  of  the  portion 
of  the  pyramid  which  lie  behind  the  plane;  that,  is  the 
portion  of  which  aaedm  is  the  buse.  It  will  be  well  tc 


PROJECTIONS.  101 

remark,  in  the  first  place,  that  removing  the  portion  in  front 
of  the  plane  of  section,  and  supposing  this  plane  transparent, 
the  interior  surfaces  of  the  pyramid  would  be  seen,  and  the 
exterior  hidden,  the  outlines  of  the  former  would  therefore  be 
drawn  full  on  the  plane  of  section,  whilst  those  of  the  latter, 
if  represented,  should  be  in  dotted  lines. 

To  construct  the  projections  of  these  lines  on  the  plane  of 
section,  we  observe  that  this  plane,  being  a  vertical  plane, 
and  O — P  being  its  trace  on  the  horizontal  plane  of  projec- 
tion, this  line  O — P  may  be  considered  as  the  ground  line  of 
these  two  planes ;  in  the  same  manner  as  G — L  is  the  ground 
line  of  the  horizontal  plane  and  the  original  vertical  plane  of 
projection.  This  being  considered,  it  is  plain  that  all  the 
parts  of  the  pyramid  should  be  projected  on  this  new  vertical 
plane  of  projection,  in  the  same  manner  as  on  the  original 
one.  Let  us  take,  for  example,  the  interior  edge,  of  which 
q — z  is  the  horizontal  projection.  The  point  q,  being  in  the 
horizontal  plane,  will  be  projected,  in  the  ground  line  O — Py 
into  q' /  and  the  point  z  would  be  projected  by  a  perpendicu- 
lar from  2  to  O — P,  at  a  height  above  O — P,  equal  to  the 
height  of  its  vertical  projection  on  the  original  vertical  plane 
above  the  ground  line  O — Z,  which  is  z' — z".  To  transfer 
these  distances  to  the  section  (Fig.  86),  take  the  distance  a — #', 
on  O — jP,  and  set  it  off  from  a'  to  q'  on  the  section ;  this  will 
give  the  projection  of  q'  on  the  section.  Next  take  the 
distance  a — z',  from  O — P,  and  set  it  off  from  a'  to  z'  on  the 
section,  and  at  z'  erect  a  perpendicular  to  the  ground  line ; 
take  from  (Fig.  85)  the  distance  z' — z",  and  set  it  off  from 
z'  to  z"  on  the  section ;  the  point  z"  will  be  the  projection  of 
the  upper  extremity  of  the  interior  edge  in  question  on  the 
plane  of  section  ;  joining  therefore  the  points  q'  and  3",  thus 
determined,  the  line  q' — z"  is  the  required  projection.  In 
like  manner,  the  projections  of  the  other  interior  and  exterior 
edges  of  the  portion  of  the  pyramid  behind  the  plane  of 
section  can  be  determined  and  drawn,  as  shown  in  the  section 
(Fig.  86). 

Remarks.  The  preceding  problems  contain  the  solutions 
of  all  cases  of  the  projections  and  sections  of  bodies,  the 
outlines  of  which  are  right  lines,  and  their  surfaces  plane 


102  INDUSTRIAL   DBAWTOG. 

figures.  As  they  embrace  a  very  large  class  of  objects  in  the 
arts,  it  is  very  important  that  these  problems  should  be 
thoroughly  understood.  One  of  the  most  useful  examples 
under  this  head  is  that  of  the  plans,  elevations,  and  sections 
of  an  ordinary  dwelling,  which  we  shall  now  proceed  to  give. 

Prob.  77.  Plans,  elevations,  <&c.,  of  a  house.  (PL  VII. 
Fig.  87.)  Let  us  suppose  the  house  of  two  stories,  with  base- 
ment and  garret  rooms.  The  exterior  walls  of  masonry, 
either  of  stone  or  brick.  The  interior  wall,  separating  the 
hall  from  the  rooms,  of  brick.  The  partition  walls  of  the 
parlors  and  basement  of  timber  frames,  filled  in  with  brick ; 
those  of  the  bedrooms  and  garret  of  timber  frames  simply. 

It  is  necessary  to  observe,  in  the  first  place,  that  the  gen- 
eral plans  are  horizontal  sections,  taken  at  some  height,  say 
one  foot,  above  the  window  sills,  for  the  purpose  of  showing 
the  openings  of  the  windows,  &c. ;  and  that  the  sections  are 
BO  taken  as  best  to  show  those  portions  not  shown  on  the 
plans,  as  the  stair- ways,  roof -framing,  &c.  In  the  second 
place,  that  in  the  plans  and  sections  are  shown  only  the  skele- 
ton, or  framework  of  the  more  solid  parts,  as  the  masonry,  or 
timber  framing  of  the  walls,  flooring,  roof,  &c. 

Plans.  Having  drawn  a  ground  line,  G — Z,  across  the 
sheet  on  which  the  drawings  are  to  be  made,  in  such  a  posi- 
tion as  to  leave  sufficient  space  on  each  side  of  it  for  the  plans 
and  elevations  respectively,  commence,  by  drawing  a  line 
A — £  parallel  to  G — Z,  and  at  a  convenient  distance  from  it 
to  leave  room  for  the  plan  of  the  first  story  towards  the 
bottom  of  the  sheet.  Take  the  line  A — B,  as  the  interior 
face  of  the  wall,  opposite  to  the  one  of  which  the  elevation  is 
to  be  represented.  Having  set  off  a  distance  on  A — B  equal 
to  the  width  between  the  side  walls,  construct  the  rectangle 
ABOD,  of  which  the  sides  A— D  and  B—C  shall  be  equal 
to  the  width  within,  between  the  front  wall  D — C  and  the 
back  A — B.  Parallel  to  these  four  sides,  draw  the  four  sides 
a — £,  b — GJ  &c.,  at  the  distance  of  the  thickness  of  the  exterior 
walls  from  them.  The  figure  thus  constructed  is  the  general 
outline  of  the  plan  of  the  exterior  walls. 

Next  proceed  to  draw  the  outline  of  the  partition  wall 
E- — F  separating  the  hall  from  the  parlors.  Next  the  walls 


PROJECTIONS.  103 

of  the  pantries,  G  and  H,  between  the  parlors.  Then  mark 
out  the  openings  of  the  windows  w,  w,  and  doors  d',d',  in  the 
walls.  Then  the  projections  of  the  fire-places,^  in  the  par- 
lors. 

Having  drawn  the  outline  of  all  these  parts  with  a  fine  ink 
line,  proceed  to  fill  it  between  the  outlines  of  the  solid  parts 
cut  through,  either  with  small  parallel  lines,  or  by  a  uniform 
black  tint.  Then  draw  the  heavy  lines  on  those  parts  from 
which  a  shadow  would  be  thrown. 

As  the  horizontal  section  will  cut  the  stairs,  it  is  usual  to 
project  on  the  plan  the  outlines  of  the  steps  below  the  plane 
in  full  lines  as  in  S  j  and,  sometimes,  to  show  the  position  of 
the  stairway  to  the  story  above,  to  project,  in  dotted  lines,  the 
steps  above  the  plane. 

If  the  scale  of  the  drawing  is  sufficiently  great  to  show  the 
parts  distinctly,  the  sections  of  the  upright  timbers  that  form 
the  framing  of  the  partition  walls  of  the  pantries  should  be 
distinguished  from  the  solid  filling  of  brick  between  them, 
by  lines  drawn  across  them  in  a  different  direction  from  those 
of  the  brick. 

The  plan  of  the  second  story  is  drawn  in  the  same  manner 
as  that  of  the  first,  and  is  usually  placed  on  one  side  of  it. 

Front  elevation.  The  figure  of  the  elevation  should  be  so 
placed  that  its  parts  will  correspond  with  those  on  that  part 
of  the  plan  to  which  it  belongs ;  that  is,  the  outlines  of  its 
walls,  of  the  doors,  windows,  &c.,  should  be  on  the  perpen- 
diculars to  the  ground  line,  drawn  from  the  corresponding 
parts  of  the  front  wall  D — C.  In  most  cases  the  outlines  of 
the  principal  lines  of  the  cornice  are  put  in,  and  those  of  the 
caps  over  the  windows,  if  of  stone  when  the  wall  is  of  brick, 
&c. ;  also  the  outline  of  the  porch  and  steps  leading  to  it. 

Section.  In  drawings  of  a  structure  of  a  simple  character 
like  this,  where  the  relations  of  the  parts  are  easily  seen,  a 
single  section  is  usually  sufficient,  and  in  such  cases  also  it  is 
usual  to  represent  on  the  same  figure  parts  of  two  different 
sections.  For  example,  suppose  O — P  to  be  the  horizontal 
trace  of  the  vertical  plane  of  section  as  far  as  P,  along  the 
hall ;  and  Q — R  that  of  one  from  the  point  Q  opposite  P 
along  the  centre  line  of  the  parlors.  On  the  first  portion  wiU 


104:  INDUSTRIAL  DBAWING. 

be  shown  the  arrangement  of  the  stairways;  and  on  the 
other  the  interior  arrangements  from  Q  towards  R.  With 
regard  to  the  position  for  the  figure  of  the  section,  it  may,  in 
some  cases,  be  drawn  by  taking  a  ground  line  parallel  to  the 
trace,  and  arranging  the  lines  of  the  figure  on  one  side  of  this 
ground  line  in  the  same  relation  to  the  side  B — (7,  of  the 
plan,  as  the  elevation  has  with  respect  to  the  side  A — B  ;  but 
as  this  method  is  not  always  convenient,  it  is  usual  to  place 
the  section  as  in  the  drawing,  having  the  same  ground  line  as 
the  elevation,  placing  the  parts  beneath  the  level  of  the 
ground  below  the  ground  line. 

For  the  better  understanding  of  the  relations  of  the  parts, 
where  the  sections  of  two  parts  are  shown  on  the  same  figure, 
it  is  well  to  draw  a  heavy  uneven  line  from  the  top  to  the 
bottom  of  the  figure,  to  indicate  the  separation  of  the  parts, 
as  in  T —  Uj  the  part  here  on  the  left  of  T — ^representing 
the  portion  belonging  to  the  hall,  that  on  the  right  the  portion 
within  the  parlors.  In  other  words,  the  figure  represents  what 
would  be  seen  by  a  person  standing  towards  the  side  A — D 
of  the  house,  were  the  portion  of  it  between  him  and  the 
plane  of  section  removed. 

This  figure  represents  the  section  of  the  stairs  and  floors, 
and  the  portion  of  the  roof  above,  and  basement  beneath  of 
the  hall ;  with  a  section  of  the  partition  walls,  floors,  roof, 
&c.,  of  the  other  portion. 

As  the  relations  of  the  parts  are  all  very  simple  and  easily 
understood,  the  parts  of  the  plans  as  well  as  of  the  elevations 
and  vertical  section  being  rectangular,  figures  of  which  all  the 
dimensions  are  put  down,  the  drawings  will  speak  for  them- 
selves better  than  any  detailed  description.  Nothing  further 
need  be  observed,  except  that  the  drawing  of  the  vertical 
section  may  be  commenced,  as  in  the  plans,  by  drawing  the 
inner  lines  of  the  walls,  and  thence  proceeding  to  put  in  the 
principal  horizontal  and  vertical  lines. 

Remarks.  In  drawings  of  tki's  class,  where  the  object  is 
simply  to  show  the  general  arrangement  of  the  structure,  the 
dimensions  of  the  parts  are  not  usually  written  on  them,  but 
are  given  by  the  scale  appended  to  the  drawing.  Where  the 
object  of  the  drawing  is  to  serve  as  a  guide  to  the  builder, 


PROJECTIONS.  105 

who  is  to  erect  the  structure,  the  dimensions  of  ever}'  part 
should  be  carefully  expressed  in  numbers,  legibly  written, 
and  in  such  a  manner  that  all  those  written  crosswise  the 
plan,  for  example,  may  read  the  same  way ;  and  those  length- 
wise in  a  similar  manner ;  in  order  to  avoid  the  inconvenience 
of  having  frequently  to  shift  the  position  of  the  sheet  to  read 
the  numbers  aright. 

Where  several  numbers  are  put  down,  expressing  the 
respective  distances  of  points  on  the  same  right  line,  it  is 
usual  to  draw  a  fine  broken  line,  and  to  write  the  numbers  on 
the  line  with  an  arrow-head  at  each  point,  as  shown  in  PI. 
XI.  Fig.  a,  which  is  read— 5  feet ;  7  feet  6  inches  ;  10  feet  9 
inches  and  7  tenths  of  an  inch. 

Where  the  whole  distance  is  also  required  to  be  set  down, 
it  may  be  done  either  by  writing  the  sum  in  numbers  over  the 
broken  line,  as  in  this  case,  23  feet  3  inches  and  7  tenths  ;  or, 
better  still,  with  the  numbers  expressing  the  partial  distances 
below  the  broken  line,  and  the  entire  distance  above  it,  as  in 
PI.  XI.  Fig.  b.  Besides  these  precautions  in  writing  the 
numbers,  each  figure  also  should  be  drawn  with  extreme 
accuracy  to  the  given  scale,  and  be  accompanied  by  an  ex- 
planatory heading,  or  reference  table. 

On  all  drawings  of  this  class  the  scale  to  which  the  drawing 
is  made  should  be  constructed  below  the  figs.,  and  be  accom- 
panied by  an  explanatory  heading ;  thus,  scale  of  inches,  of 
feet  to  inches,  or  feet.  In  some  cases,  the  draughtsman  will 
find  it  more  convenient  to  construct  a  scale  on  a  strip  of  draw- 
ing paper  for  the  drawing  to  be  made,  than  to  use  the  ivory 
one ;  particularly,  as  with  the  paper  one  he  can  lay  down  his 
distances  at  once,  without  first  taking  them  off  with  the 
dividers,  in  the  same  way  as  points  are  transferred  by  a  strip 
of  paper.  In  either  case,  the  scale  should  be  so  divided  as  to 
aid  in  reading  and  setting  off  readily  any  required  distance. 
The  best  mode  of  division  for  this  purpose  is  the  decimal ; 
and  the  following  manner  of  constructing  the  scale  the  most 
convenient: — Having  drawn  a  right  line,  set  off  accurately 
from  a  point  at  its  left-hand  extremity,  ten  of  the  units  of 
the  required  scale,  and  number  these  from  the  left  10,  9,  &c., 
to  0.  From  the  0  point,  set  off  on  the  right,  as  many  equal 


106  INDU8TEIAJL   DRAWING. 

distances,  each  of  the  length  of  the  part  from  0  to  10,  as  may 
be  requisite,  and  number  these  from  0  to  the  right  10,  20, 
&c.  From  this  scale  any  number  of  tens  and  units  can  be  at 
once  set  off.  This  scale  should  be  long  enough  to  set  off  the 
longest  dimension  on  the  drawing. 

See,  for  example,  the  scale  and  its  heading  at  bottom  of  PL 

m 

Preliminary  Problems  in  Projections. 

Before  entering  upon  the  drawings  of  some  of  the  many 
objects  belonging  to  this  class,  it  will  be  necessary  to  show 
the  manner  of  making  the  projections  of  the  cone,  cylinder, 
and  sphere  ;  that  of  obtaining  their  intersections  by  a  plane ; 
and  also  that  of  representing  their  intersections  with  each 
other. 

Cylinder.  (PL  X.  Fig.  107.)  If  we  suppose  a  rectangle, 
ABCD,  cut  out  of  any  thin  inflexible  material,  as  stiff  paste- 
board, tin,  &c.,  and  on  it  a  line,  0 — P,  drawn  through  its 
centre,  parallel  to  its  side  B — O,  for  example ;  this  line  being 
so  fixed  that  the  rectangle  can  be  revolved,  or  turned  about 
0— P,  it  is  clear  that  the  sides  A— D,  and  B—  C\  will,  in 
every  position  given  to  the  rectangle,  be  still  parallel  to 
O — P,  and  at  the  same  distance  from  it.  The  side  B — C,  or 
A — Z>,  therefore,  may  be  said  to  describe  or  generate  a 
surface,  in  thus  revolving  about  O — P,  on  which  right  lines 
can  be  drawn  parallel  to  O — P.  It  will  moreover  be  ob- 
served, that  as  the  points  A  and  B,  with  D  and  C  are, 
respectively,  at  the  same  perpendicular  distance  (each  equal 
to  O — B)  from  the  axis  O — P,  they,  in  revolving  round  it, 
will  describe  the  circumferences  of  circles,  the  radii  of  which 
will  also  be  equal  to  0 — B.  In  like  manner,  if  we  draw  any 
other  line  parallel  to  A — B,  or  D — C,  as  a — 5,  its  point  5,  or 
«,  will  describe  a  circumference  having  for  its  radius  o — J, 
which  is  also  equal  to  O — B. 

The  surface  thus  described  is  termed  a  right  circular 
cylinder,  and,  from  the  mode  of  its  generation,  the  following 
properties  may  be  noted ; — 1st,  any  line  drawn  on  it  parallel 
to  the  line  O — JP,  which  last  is  termed  the  axis,  is  a  right  line. 


PROJECTIONS.  107 

and  is  termed  a  right  line  element  of  the  cylinder ;  2d,  any 
plane  passed  through  the  axis  will  cut  out  of  the  surface  two 
right  line  elements  opposite  to  each  other;  3d,  as  the  planes 
of  the  circumferences  described  by  the  points  Z?,  £,  &c.,  are 
perpendicular  to  the  axis,  any  plane  so  passed  will  cut  out 
of  the  surface  a  circumference  equal  to  those  already  de- 
scribed. 

Prob.  85.  (PI.  X.  Fig.  108.)  Projections  of  the  cylinder. 
Let  us  in  the  first  place  suppose  the  cylinder  placed  on  the 
horizontal  plane  with  its  axis  vertical.  In  this  position  all 
of  its  right  line  elements,  being  parallel  to  the  axis,  will  also 
be  vertical  and  be  projected  on  the  horizontal  plane  in  points, 
at  equal  distances  from  the  point  0,  in  which  the  axis  is  pro- 
jected. Describing  therefore  a  circle,  from  the  point  0,  with 
the  radius  equal  to  the  distance  between  the  axis  and  the  ele- 
ments, this  circle  will  be  the  horizontal  projection  of  the  sur- 
face of  the  cylinder. 

To  construct  the  vertical  projection,  let  us  suppose  the 
generating  rectangle  placed  parallel  to  the  vertical  plane  ;  in 
this  position,  it  will  be  projected  into  the  diameter  a — b  of 
the  circle  on  the  horizontal  plane,  and  into  the  equal  rectan- 
gle a'b''(}'df  on  the  vertical  plane ;  in  which  last  figure  c' — d' 
will  be  the  projection  of  circle  of  the  base  of  the  cylinder; 
the  line  a' — b'  that  of  its  top ;  and  the  lines  a' — d',  and  br — cf 
those  of  the  two  opposite  elements.  As  all  the  other  elements 
will  be  projected  on  the  vertical  plane  between  these  two  last, 
the  rectangle  a'b'c'd'  will  represent  the  vertical  projection  of 
the  entire  surface  of  the  cylinder.  To  show  this  more  clearly, 
let  us  suppose  the  generating  rectangle  brought  into  a  position 
oblique  to  the  vertical  plane,  as  e—f  for  example ;  in  this 
position,  the  two  elements  projected  in  e,  and  ./on  the  hori- 
zontal plane,  would  be  projected  respectively  into  e' — o',  and 
f — A',  on  the  vertical  plane ;  and  as  the  one  projected  in  f, 
f — A',  lies  behind  the  cylinder,  it  would  be  represented  by  a 
dotted  line. 

All  sections  of  the  cylinder  parallel  to  the  base,  or  perpen- 
dicular to  the  axis,  being  circles,  equal  to  that  of  the  base, 
will  be  projected  on  the  horizontal  plane  into  the  circle  of  the 
base,  and  on  the  vertical  in  lines  parallel  to  c' — d'.  All  sec- 


108  INDUSTRIAL   DRAWING. 

tions  through  the  axis  will  be  equal  rectangles;  which,  in 
horizontal  projection,  will  be  diameters  of  the  circle,  as  e—f; 
and,  in  vertical  projection,  rectangles  as  e'f'h'o'.  All  sections, 
as  3f — JVi  parallel  to  the  axis,  will  cut  out  two  elements  pro- 
jected respectively  in  m,m'—p';  and  n,  n' — u'. 

Note.  The  manner  of  constructing  oblique  sections  will 
be  given  farther  on. 

Prob.  86.  (PL  X.  Fig.  109.)  Cone.  If  in  a  triangle, 
having  two  equal  sides  d' — -p\  and  tf—p',  we  draw  a  line 
p' — 0',  bisecting  the  base  cf — <#',  and  suppose  the  triangle 
revolved  about  this  line  as  an  axis,  the  equal  sides  will 
describe  the  curved  surface,  and  the  base  the  circular  base 
of  a  body  termed  a  right  cone  with  a  circular  base. 

From  what  has  been  said  on  the  cylinder,  it  will  readily 
appear  that,  in  this  case,  any  plane  passed  through  the  axis 
will  cut  out  of  the  curved  surface  two  right  lines  like  d'—p', 
and  c'—p';  and,  therefore,  that  from  the  pointy/  of  the  sur- 
face, termed  the  vertex,  right  lines  can  be  drawn  to  every 
point  of  the  circumference  of  the  base.  These  right  lines  are 
termed  the  right  line  elements  of  the  cone.  It  will  also  be 
equally  evident  that  any  line,  as  a — 1>,  drawn  perpendicular 
to  the  axis,  will  describe  a  circle  parallel  to  the  base,  of  which 
a — b  is  the  diameter,  and,  therefore,  that  every  section  perpen- 
dicular to  the  axis  is  a  circle. 

From  these  preliminaries,  the  manner  of  projecting  a  right 
cone  with  a  circular  base,  and  its  sections  either  through  the 
vertex,  or  perpendicular  to  the  axis,  will  be  readily  gathered 
by  reference  to  the  figures. 

The  horizontal  projection  of  the  entire  surface  will  be 
within  the  circumference  of  the  base,  that  of  the  vertex  being 
at  p.  The  vertical  projection  of  the  surface  will  be  the 
triangle  d'p'd  /  as  the  projection  of  all  the  elements  must 
lie  within  it.  Any  plane  of  section  through  the  axis^  as  0—f, 
will  cut  from  the  surface  two  elements  projected  horizontally 
mp—fandjp — e;  and  vertically  in  e'  — p'  and/'— -p'.  Any 
plane  passed  through  the  vertex,  of  which  M—  ^Vis  the  hori- 
zontal trace,  will  cut  from  the  surface  two  right  line  elements, 
of  which  % — m  and  p — n  are  the  horizontal,  and  p' — ra'  and 
p' — n'  the  vertical  projections.  Any  plane  as  R — S  passed 


PROJECTIONS.  109 

perpendicular  to  the  axis  will  cut  out  a  circle,  of  winch  a' — V 
is  the  vertical,  and  the  circle  described  from  p,  with  a  diame- 
ter a — b  equal  to  a' — b',  is  the  horizontal  projection. 

Prob.  87.  Sphere.  (PI.  X.  Fig.  110.)  Drawing  any 
diameter,  O — jP,  in  a  circle,  and  revolving  the  figure  around 
it,  the  circumference  will  describe  the  surface  of  a  sphere ; 
the  diameter  A — B  perpendicular  to  the  axis  will  describe  a 
circle  equal  to  the  given  one ;  and  any  chord,  as  a — b,  will 
describe  a  circle,  of  which  a—b  is  the  diameter.  Supposing 
the  sphere  to  rest  on  the  horizontal  plane,  having  the  axis 
O — P  vertical,  every  section  of  the  sphere  perpendicular  to 
this  axis  will  be  projected  on  the  horizontal  plane  in  a  circle ; 
and,  as  the  section  through  the  centre  cuts  out  the  greatest 
circle,  the  entire  surface  will  be  projected  within  the  circle, 
the  diameter  of  which,  A — B,  is  equal  to  the  diameter  of  the 
generating  circle.  In  like  manner  the  vertical  projection 
will  be  a  circle  equal  to  this  last.  The  centre  of  the  sphere 
will  be  projected  in  C  and  C'  /  and  the  vertical  axis  in  C 
and  O — P.  Any  section,  as  R — /S,  perpendicular  to  the 
axis,  will  cut  out  a  circle  of  which  a — b  is  the  diameter,  and 
also  the  vertical  projection ;  the  horizontal  projection  will  be 
a  circle  described  from  C  with  a  diameter  equal  to  a — b. 
Any  section,  as  M — jW,  by  a  vertical  plane  parallel  to  the 
vertical  plane  of  projection,  will  also  be  a  circle  projected  on 
the  horizontal  plane  in  m — n  j  and  on  the  vertical  plane  in  a 
circle,  described  from  C'  as  a  centre,  and  with  a  diameter 
equal  to  m — n. 

Remarks.  The  three  surfaces  just  described  belong  to  a 
class  termed  surfaces  of  revolution^  which  comprises  a  very 
large  number  of  objects ;  as  for  example,  all  those  which  are 
fashioned  by  the  ordinary  turner's  lathe.  They  have  all 
certain  properties  in  common,  which  are — 1st,  that  all  sec- 
tions perpendicular  to  the  axis  of  revolution  are  circles ; 
2d,  that  all  sections  through  the  axis  of  revolution  are  equal 
figures. 

Prob.  88.  (PI.  X.  Fig.  111.)  To  construct  the  section  of 
a  right  circular  cylinder  by  a  plane  oblique  to  its  axis,  and 
perpendicular  to  the  vertical  plane  of  projection. 

1st  Method.    Let  aobh  and  a'Vc'd'  be  the  projections  of 


110  INDUSTRIAL   DBA  WING. 

the  cylinder,  and  M—N  the  vertical  trace  of  the  plane  of 
section.  This  plane  will  cut  the  surface  of  the  cylinder  in 
the  curve  of  an  ellipse,  all  the  points  of  which,  since  the  curve 
is  on  the  surface  of  the  cylinder,  will  be  horizontally  projected 
in  the  circle  aolh,  and  vertically  in  m' — n',  since  the  curve 
lies  also  in  the  plane  of  section.  Now  the  plane  of  section 
cuts  the  two  elements,  projected  in  a,  a' — d',  and  t>,  ~b' — c',  in 
the  points  projected  in  a,  and  m' ;  and  b,  and  ri.  It  cuts  the 
two  elements  projected  in  o' — £',  and  0,  and  h,  in  the  points 
projected  vertically  in  r',  and  horizontally  in  o  and  h.  Tak- 
ing any  other  element  as  the  one  projected  in  0,  e' — a?',  the 
plane  cuts  it  in  a  point  projected  vertically  in  q',  and  horizon- 
tally in  e.  The  same  projection  e  and  q'  would  also  corre- 
spond to  the  point  in  which  the  element  projected  in^  e' — a?', 
is  cut  by  the  plane ;  and  so  on  for  any  other  pair  of  elements 
similarly  placed,  with  respect  to  the  line  a — b  on  the  base. 

%d  Method.  As  the  plane  of  section  is  perpendicular  to 
the  vertical  plane  of  projection,  its  trace  on  the  horizontal 
plane  (PL  YI.  Fig.  82)  will  be  the  line  M—P,  perpendicular 
to  the  ground  line.  In  like  manner,  if  the  plane  of  section 
was  cut  by  another  horizontal  plane,  as  g — I  j  or  if  the  origi- 
nal horizontal  plane  was  moved  up  into  the  position  of  the 
second,  g — I  would  become  the  new  ground  line,  and  the  line 
p—p",  the  trace  of  the  plane  of  section  on  this  second  hori- 
zontal plane.  From  this  it  is  clear,  that  the  line  in  which  the 
second  horizontal  plane  cuts  the  plane  of  section  is  projected 
vertically  in  the  point  p\  and  horizontally  in  the  line  p — p", 
on  the  first  horizontal  plane.  But  the  horizontal  plane  of 
which  g — I  is  the  vertical  trace,  being  perpendicular  to  the 
axis  of  revolution,  cuts  out  of  the  cylinder  a  circle  which  is 
projected  into  aobfi  /  and  as  it  is  equally  clear  that  since  the 
line,  projected  in  p',p — -p",  lies  in  the  horizontal  plane  g — I, 
it  must  cut  the  circle  also,  the  points  i  and  w,  in  which  p — -p" 
cuts  the  circle  of  the  base,  will  thus  be  the  projections  of 
the  points  in  question.  A  like  reasoning  and  corresponding 
construction  would  hold  true  for  any  other  points,  as  r',  q', 
&c. 

Remarks.  Of  the  two  methods  here  given,  the  2d,  as  will 
be  seen  in  what  follows,  is  the  more  generally  applicable,  as 


PEOJEOTION8.  Ill 

requiring  more  simple  constructions ;  but  the  1st  is  the  more 
suitable  for  this  particular  case,  as  giving  the  results  by  the 
simplest  constructions  that  the  problem  admits  of. 

Prob.  89.  (PI.  X.  Fig.  112.)  To  construct  the  sections 
of  a  rig ht  cone  with  a  circular  base  by  planes  perpendicular 
to  the  vertical  plane  of  projection. 

This  problem  comprises  five  cases.  1st,  where  the  plane 
of  section  is  perpendicular  to  the  axis ;  2d,  where  it  passes 
through  the  vertex  of  the  cone ;  3d,  where  its  vertical  trace 
makes  a  smaller  angle  with  the  ground  line  than  the  vertical 
projection  of  the  adjacent  element  of  the  cone  does;  4rth, 
where  the  trace  makes  the  same  angle  as  the  projection  of 
the  adjacent  element  with  the  ground  line,  or,  in  other  words, 
is  parallel  to  this  projection ;  5th,  where  the  trace  makes  a 
greater  angle  than  this  line  with  the  ground  line. 

Cases  1st  and  2d  have  already  been  given  in  the  projections 
of  the  cone  (PI.  X.  Fig.  109)  ;  the  remaining  three  alone  re- 
main to  be  treated  of. 

Case  3d.  In  this  case,  the  angle  NML,  between  the  trace 
M—  ^Tof  the  plane  and  the  ground  line  G — Z,  is  less  than 
that  o'a'b',  or  that  between  the  same  line  and  the  element 
projected  in  o' — a'/  and  the  curve  cut  from  the  surface  of 
the  cone  will  be  an  ellipse. 

From  what  precedes  (Prob.  86),  if  the  cone  be  intersected 
by  a  horizontal  plane,  of  which  g — I  is  the  trace,  it  will  cut 
from  the  surface  a  circle  of  which  c' — d'  is  the  diameter,  and 
which  will  be  projected  on  the  horizontal  plane  in  the  cir- 
cumference described  from  o  on  this  diameter;  this  plane 
will  also  cut  from  the  plane  of  section  a  right  line,  projected 
vertically  in  r',  and  horizontally  in  r — r"  /  and  the  projec- 
tions of  the  points  8  and  -w,  in  which  this  line  cuts  the  projec- 
tion of  the  circumference,  will  be  two  points  in  the  projection 
of  the  curve  cut  from  the  surface. 

Constructing  thus  any  number  of  horizontal  sections,  as 
g — I,  between  the  points  ra'  and  n'  (the  vertical  projections 
of  the  points  in  which  the  plane  of  section  cuts  the  two  ele- 
ments parallel  to  the  vertical  plane),  any  number  of  points  in 
the  horizontal  projection  cf  the  curve  can  be  obtained  like 
the  two,  s  and  u,  already  found. 


112  INDUSTEIAL  DEAWINQ. 

On  examining  the  horizontal  projection  of  the  curve  it  will 
be  seen  that  each  pair  of  points,  like  s  and  u,  are  on  lines 
perpendicular  to  m — n,  and  at  the  same  distance  from  it ; 
and  that  the  points  m  and  n  are  on  the  line  a — b,  into  which 
the  two  elements  parallel  to  the  vertical  plane  are  projected 
It  will  also  be  seen  that  the  two  points  s  and  u,  determined 
by  that  horizontal  plane  which  bisects  the  line  m' — n',  are 
the  farthest  from  the  line  m — n.  The  line  m — n,  which  is 
the  horizontal  projection  of  the  longest  line  of  the  curve  of 
section,  is  the  transverse  axis  of  the  ellipse  into  which  this 
curve  is  projected,  and  the  line  * — u  the  conjugate  axis. 

Prob.  90.  Cases  4  and  5.  (PI.  X.  Fig.  113.)  Although 
the  curve  cut  from  the  surface  in  each  of  these  cases  is  dif- 
ferent, Case  4:  giving  a  parabola,  and  Case  5  an  hyperbola, 
the  manner  of  determining  the  projections  of  the  curve  is  the 
same ;  and  but  one  example  therefore  will  be  requisite  to 
illustrate  the  two  cases. 

Taking  the  vertical  trace  M — JV",  of  the  plane  of  section, 
parallel  to  a! — <?',  its  horizontal  trace,  M- — P,  will  cut  the 
circle  of  the  base  in  two  points,  n  and  v,  which  are  points  of 
the  horizontal  projection  of  the  curve  of  section.  The  plane 
cuts  the  element  parallel  to  the  vertical  plane,  and  which  is 
projected  in  o — b,  and  o' — J',  in  a  point,  projected  in  m  and 
vn! .  The  three  points  thus  found  are  the  extreme  points  of 
the  horizontal  projection  required.  To  find  the  intermediate 
points,  proceed  as  in  Case  3,  by  intersecting  the  cone  and 
plane  of  section  by  horizontal  planes,  as  g — Z,  and  then  deter- 
mining the  projections  of  the  points,  as  s  and  u,  in  which  the 
lines  cut  from  the  plane  intersect  the  circumferences  cut  from 
the  surface  of  the  cone.  A  curve  drawn  through  the  points 
thus  found  will  be  the  required  projection. 

Prob.  91.  (PI.  XL  Fig.  114.)  To  find  the  projections  of 
the  curve  of  intersection  cut  from  a  sphere  by  a  plane  of  sec- 
tion, as  in  the  preceding  problems. 

Let  o,  o'  be  the  projections  of  the  centre  of  the  sphere; 
M- — JVand  M- — P  the  traces  of  the  plane  of  section. 

Intersecting  the  sphere  and  plane  by  horizontal  planes  like 
g — Z,  as  in  the  preceding  problems,  and  finding  the  projec- 
tions of  the  lines  cut  from  them,  the  points,  as  s  and  w,  in 


PROJECTIONS.  113 

which  these  intersect,  will  be  points  in  the  projection  of  the 
required  curve. 

The  points  m'  and  n',  in  which  M- — N~  cuts  the  vertical 
projection  of  the  sphere,  will  be  the  vertical  projections  of 
the  points  in  which  the  plane  cuts  the  circle  parallel  to  the 
vertical  plane,  and  which  is  projected  in  a — &/  the  horizon- 
tal projections  of  these  points  will  be  m,  and  n,  on  a — J. 
The  horizontal  projections  of  the  points  in  which  the  plane 
cuts  the  horizontal  plane  g' — I',  through  the  centre  of  the 
sphere,  are  c  and  d.  The  projection  of  the  curve  of  intersec- 
tion is  an  ellipse,  of  which  c — d  is  the  transverse,  and  m — n 
the  conjugate  axis. 

Fig.  A  is  the  circle  cut  from  the  sphere,  the  diameter  of 
which  is  m! — n'. 

Prob.  92.  (PL  X.  Fig.  115.)  To  construct  the  curves  of 
intersection  of  the  planes  of  section  and  the  surf  aces  in  the 
preceding  problems. 

In  each  of  the  preceding  problems,  the  horizontal  and  ver- 
tical projections  of  the  curves  cut  from  the  surface  have  alone 
been  found ;  the  true  dimensions  of  these  curves,  as  they  lie 
in  their  respective  planes  of  section,  remain  to  be  deter- 
mined. 

1st  Method.  An  examination  of  (Figs.  Ill,  112,  113, 114) 
will  show  that  as  the  points  vertically  projected  in  m'  and  n' 
and  horizontally  on  the  lines  a — 5,  are  at  the  same  distance 
from  the  vertical  plane  of  projection,  the  line  joining  them 
will  be  vertically  projected  into  its  true  length  m' — n'.  In 
like  manner,  the  lines  cut  from  the  plane  of  section  by  the 
horizontal  planes,  as  g — I,  and  projected  horizontally  in  s — u, 
are  projected  in  their  true  length,  and  they,  moreover,  lie  in 
the  plane  of  section  and  are  perpendicular  to  the  line  m' — n'. 
If  we  construct  therefore  the  true  positions  of  these  lines, 
with  respect  to  each  other,  we  shall  obtain  the  true  dimen- 
sions of  the  curve.  To  do  this,  draw  any  line,  and  set  off 
upon  it  a  distance  m! — n'  equal  to  mf — n' /  from  m',  set  off 
the  distances  mf — q',  m! — /•',  m' — j/,  &c.,  equal  to  those  on 
the  corresponding  projections  m'  —  q',  &c. ;  through  the 
points  q',  r',  &c.,  draw  perpend iculars  to  m' — n' ;  and  on 
these  last,  from  the  same  points,  set  off  each  way  the  equal 
8 


114  INDUSTRY  AT.   DRAWING. 

distances  r' — 8,  r' — u,  &c.,  the  half  of  the  corresponding 
lengths  5 — u,  &e.  Through  the  points  m',  s'y  &c.,  thus  set 
off,  draw  a  curved  line ;  this  curve  will  be  equal  to  the  curve 
cut  from  the  surface. 

Remarks,  This  curve  in  the  cylinder,  and  in  Case  3  of  the 
cone,  is  an  ellipse,  of  which  m' — n'  is  the  length  of  the  trans- 
verse axis,  and  the  line  s — u,  corresponding  to  the  point  r'y 
the  centre  of  m' — n' ',  is  the  conjugate  axis.  In  the  last  two 
Oases  of  the  cone,  the  curve  is  either  a  parabola,  or  an  hyper- 
bola ;  and  in  the  sphere  it  is  in  all  cases  a  circle. 

Prob.  93.  Zd  Method.  (PI.  X.  Figs.  112,  113.)  Going 
back  to  what  has  been  shown,  in  describing  the  manner  of 
generating  any  surface  of  revolution,  we  find  that  every  point 
in  a  plane,  revolved  about  an  axis  in  that  plane,  revolves  in 
the  circumference  of  a  circle,  the  radius  of  which  is  the  per- 
pendicular drawn  from  the  point  to  the  axis.  If  then  we 
know,  or  can  find  the  perpendicular  from  a  point  in  a  plane 
to  a  line  in  the  same,  taken  as  an  axis  of  revolution,  we  can 
always  construct  the  position  of  this  point,  in  any  given  posi- 
tion of  the  plane.  The  case  of  points  in  the  vertical  plane 
of  projection,  is  already  a  familiar  illustration  of  this  prin- 
ciple. The  distances  of  these  points  from  the  ground  line 
being  the  same  after  the  vertical  plane  is  revolved  around  it, 
to  coincide  with  the  horizontal  plane,  as  they  were  before  the 
revolution. 

Now,  to  apply  this  principle  to  finding  the  true  dimensions 
of  a  curve  of  section  ;  let  us  revolve  the  plane  of  section,  in 
which  the  curve  lies,  around  its  vertical  trace  M- — N,  for 
example,  until  the  plane  of  section  is  brought  to  coincide  with 
the  vertical  plane  of  projection.  The  point  of  the  curve,  of 
which  ra  and  m',  for  example,  are  the  projections,  is  at  a 
perpendicular  distance  from  the  vertical  plane  equal  to  the 
distance  m — m"  of  its  horizontal  projection  from  the  ground 
line;  but  as  the  vertical  projection  of  the  point  is  on  the 
trace  M — N,  the  length  m — m"  is  also  the  perpendicular 
distance  of  the  point  from  M — N~.  Drawing  then,  from  m'. 
&  perpendicular  to  M- — N,  and  setting  off  upon  it  a  distance 
m' — m'"  equal  to  m" — m,  the  point  in'"  will  be  the  position 
of  the  point  of  the  curve,  when  its  plane  is  revolved  around 


PROJECTIONS.  115 

M—N,  to  coincide  with  the  vertical  plane.  In  like  manner, 
the  positions  of  the  points  projected  in  s  and  u  are  found,  by 
drawing  through  the  point  r',  their  vertical  projection,  a 
perpendicular  to  M—N~,  and  setting  off  on  it  the  lengths 
r' — s,  and  /•' — u,  equal  to  the  respective  lengths  r — s,  and 
r — u,  the  distances  of  the  horizontal  projections  of  the  points 
from  the  ground  line ;  and  so  on  for  the  other  points. 

Remarks.  A  moment's  examination  of  the  two  methods, 
just  explained,  will  show  that  their  results  are  identical,  the 
lines,  as  s — u,  for  example,  being  of  the  same  length  and 
occupying  the  same  position,  with  respect  to  the  equal  lines 
m' — n',  and  m!" — n'".  The  second  shows  the  connection 
between  the  different  points  more  clearly  than  the  first,  and  is, 
in  most  cases,  the  more  convenient  one  for  constructing  them. 

It  will  be  seen  farther  on,  that  this  principle  of  finding  the 
true  positions  of  points  in  a  plane,  with  respect  to  each  other, 
by  revolving  the  plane  around  some  line  in  it,  selected  as  an 
axis,  is  of  frequent  and  convenient  application  in  obtaining 
the  projections  of  points. 

Projections  of  the  right  cylinder  and  right  cone  in  posi- 
tions where  their  axes  are  oblique  to  either  one  or  both  planes 
of  projection. 

Preliminary  Problems.  The  2d  method  employed  in  the 
preceding  proposition,  by  which  the  true  positions  of  any 
points  in  a  plane  are  found,  when  their  projections  are  given, 
by  revolving  the  plane  to  coincide  with  either  the  vertical  or 
horizontal  plane  of  projection,  may  be  used,  with  great 
advantage,  for  constructing  the  projections  of  any  series  of 
points  contained  in  a  plane,  in  any  assumed  position  of  this 
plane,  when  the  projections  of  the  same  points  are  known  in 
any  given  position  of  the  plane.  ,  , 

Prob.  94  (PI.  X.  Fig.  116.)  Let  a'b'c'd'  be  the  vertical 
projection  of  a  circle,  contained  in  a  plane  parallel  to  the 
vertical  plane  of  projection,  of  which  O, — P  is  the  horizontal 
trace.  As  the  plane,  containing  the  circle,  is  perpendicular 
to  the  horizontal  plane,  all  the  points  of  the  circle  will  be 
projected  horizontally  in  the  trace  0 — P.  The  points,  of 
which  a'  and  b',  for  example,  are  the  vertical  projections, 
being  projected  in  a  and  b}  &c. 


116  INDUSTRIAL   DRAWING. 

Having  the  projections  of  the  circle  in  this  position  of  its 
plane,  let  it  be  required  to  find  the  projections  when  the 
plane  is  revolved  about  any  vertical  line  drawn  in  it,  as,  for 
example,  the  one  of  which  M — N  is  the  vertical,  and  O  the 
horizontal  projection,  and  brought  into  a  position,  as  0 — P', 
oblique  to  the  vertical  plane  of  projection. 

From  a  slight  examination  of  what  takes  place  in  this 
revolution,  it  will  be  clearly  seen,  that  the  positions  of  the 
points  in  the  circle,  with  respect  to  the  horizontal  plane  of 
projection,  are  not  changed,  and  that  they  are  only  moved  in 
the  revolution  farther  from  the  vertical  plane.  As  a  familiar 
illustration  of  this,  suppose  a  circle  described  on  the  surface 
of  an  ordinary  door,  when  closed.  When  the  door  is  opened, 
all  the  points  of  the  circle  will  still  be  at  the  same  distances 
above  the  floor,  considered  as  a  horizontal  plane,  as  they  were 
when  the  door  was  closed,  the  only  change  being  in  their 
farther  removal  from  the  surface  of  the  wall,  considered  as  a 
vertical  plane.  Let  O — P'  then  be  the  trace  of  the  plane 
containing  the  circle  in  its  new  assumed  position,  the  points 
in  horizontal  projection  a,  c,  b,  &c.,  will,  in  this  new  position, 
be  found  at  a^,  c^  b^  at  the  same  distances  O — «15  from  (9,  as  in 
their  first  positions  O — a,  &c.  The  vertical  projections  of 
these  points  in  their  new  positions  will  be  found,  by  drawing 
through  the  points  «l5  &c.,  perpendiculars  to  the  ground  line, 
and  setting  off  on  these  the  same  distances  above  the  ground 
line,  as  the  original  vertical  projections  of  the  points.  Describ- 
ing a  curve  through  the  points  a!'c"b"d",  &c.,  corresponding 
to  a',  b',  c',  d',  &c.,  thus  obtained,  it  will  be  the  vertical  pro- 
jection of  the  circle  required.  This  curve  will  be  an  ellipse, 
of  which  the  line  c" — d",  equal  to  c' — d',  is  the  transverse 
axis;  and  a" — b",  the  vertical  projection  of  the  diameter 
a' — V  parallel  to  the  horizontal  plane,  is  the  conjugate  axis. 

Remarks.  From  an  examination  of  the  circle  and  the 
ellipse,  it  will  be  seen,  that  the  diameter  of  the  circle  <?' — d', 
which,  in  the  revolution  of  the  plane,  remains  parallel  to  the 
vertical  plane,  becomes  the  transverse  axis  of  the  ellipse, 
whilst  the  one,  a' — 5',  perpendicular  to  c' — d ',  and  which,  in 
the  revolution,  changes  its  position  to  the  vertical  plane,  be- 
comes the  conjugate  axis.  Also  that  all  the  lines  of  the  circle 


PROJECTIONS.  Ill 

parallel  to  c' — d',  as  m' — m",  remain  of  the  same  length  on 
the  ellipse,  and  parallel  to  o" — d". 

In  all  oblique  positions  that  can  be  given  to  the  plane  of 
a  circle,  with  respect  to  either  plane  of  projection,  the  pro- 
jection of  the  circle,  upon  that  plane  of  projection  to  which 
it  is  oblique,  will  be  an  ellipse  ;  the  transverse  axis  of  which 
will  be  that  diameter  of  the  circle  which  is  parallel  to  the 
plane  of  projection,  and  the  conjugate  axis  is  the  projection 
of  that  diameter  which  is  perpendicular  to  the  transverse  axis. 
Whenever,  therefore,  the  projections  of  these  two  diameters 
can  be  found,  the  curve  of  the  ellipse  can  be  described  by  the 
usual  methods. 

Prob.  95.  (PL  X.  Fig.  116.)  Having  the  projections  of 
a  figure  contained  in  a  plane  perpendicular  to  one  plane  of 
projection,  but  oblique  to  the  other,  to  find  the  true  dimen- 
sions of  the  figure. 

This  problem  which  is  the  converse  of  the  preceding  is  also 
of  frequent  application,  in  finding  the  projections  of  points, 
as  will  be  seen  further  on. 

Suppose  0 — P'9  the  horizontal  trace  of  a  plane  perpen- 
dicular to  the  horizontal  plane  of  projection,  but  oblique  to 
the  vertical ;  and  let  the  points  a1}  cl5  bv,  &c.,  be  the  horizontal 
projections,  and  a",  c",  b",  d"  the  vertical  projections  of  the 
points  of  a  figure  contained  in  this  plane.  If  we  suppose  the 
plane  to  be  revolved  about  any  line  drawn  in  it  perpendicular 
to  the  horizontal  plane,  as  the  one  of  which  O,  and  M- — JY 
are  the  projections,  until  it  is  brought  parallel  to  the  vertical 
plane,  the  horizontal  trace,  after  revolution,  will  be  found  in 
the  position  O — P,  parallel  to  the  ground  line ;  and  the 
points  Oi'  (\r,  &c.,  in  the  new  positions  a,  c,  &c.,  on  0 — P. 
The  vertical  projections  of  these  points  will  be  found  in 
a',  c',  b',  d',  &c.,  at  the  same  height  above  the  ground  line  as 
in  their  primitive  positions.  The  new  figure,  being  parallel 
to  the  vertical  plane,  will  be  projected  in  one  a'Vcd',  &c., 
equal  to  itself. 

Remarks.  This  method,  it  may  be  well  to  note,  will  also 
serve  to  find  the  projections  of  any  points,  or  of  a  figure,  con- 
tained in  a  plane  perpendicular  to  the  horizontal,  but  oblique 
to  the  vertical  plane,  for  any  new  oblique  position  of  this 


118  INDUSTRIAL  DRAWING. 

plane,  with  respect  to  the  vertical  plane,  taken  up,  by  revolv- 
ing it  around  a  line  assumed  in  a  like  position  to  the  one  of 
which  the  projections  are  O,  and  M- — N. 

Prob.  96.  (PI.  X.  Fig.  117.)  To  construct  the  projections 
of  a  given  right  cylinder  with  a  circular  base,  the  cylinder 
resting  on  the  horizontal  plane,  with  its  axis  oblique  to  the 
vertical  plane  of  projection. 

We  have  seen  (Prob.  85)  that  the  axis  of  a  right  cylinder 
is  perpendicular  to  the  planes  of  the  circles  of  its  ends ;  and 
that,  when  the  axis  is  parallel  to  one  of  the  planes  of  projec- 
tion, the  projection  of  the  cylinder  on  that  plane  is  a  rectan- 
gle, the  length  of  which  is  equal  to  the  length  of  the  axis, 
and  the  breadth  equal  to  the  diameter  of  the  circles  of  its 
ends  ;  constructing,  therefore,  a .  rectangle  abed,  of  which  the 
side  b — c  is  equal  to  the  length  of  the  axis,  and  the  side  a — •b 
is  the  diameter  of  the  circle  of  the  ends,  this  figure  will  be 
the  horizontal  projection  of  the  cylinder;  the  line  o—p, 
drawn  bisecting  the  opposite  sides  a — b,  c—d,  is  the  projec- 
tion of  the  axis. 

To  make  the  vertical  projection,  it  will  be  observed,  that 
the  bottom  element  of  the  cylinder,  being  on  the  horizontal 
plane,  and  projected  in  the  line  o—p,  will  be  projected  into 
the  ground  line,  in  the  line  o"—p"  ;  whilst  the  top  element, 
which  is  also  horizontally  projected  in  o—p,  will  be  projected 
in  o'—pr,  at  a  height  above  the  ground  line  equal  to  the 
diameter  of  the  cylinder.  The  line  s' — t*,  parallel  to  o"—p", 
and  bisecting  the  opposite  sides  o' — 0",  and  p' — -p",  is  the 
vertical  projection  of  the  axis. 

The  planes  of  the  circles  of  the  two  ends,  being  perpendi- 
cular to  the  horizontal  plane,  but  oblique  to  the  vertical,  the 
circles  (Prob.  94)  will  be  projected  into  ellipses  on  the  verti- 
cal plane ;  the  transverse  axes  of  which  will  be  the  lines 
o' — o",  and  p' — -p",  the  vertical  projections  of  the  diameters 
of  the  circles  parallel  to  the  vertical  plane ;  and  the  lines 
a' — b',  and  df — c',  the  vertical  projections  of  the  diameters 
parallel  to  the  horizontal  plane,  are  the  conjugate  axes 
Constructing  the  two  ellipses  a'o"b'o',  and  d'p"c'p',  the  ver- 
tical projection  of  the  cylinder  will  be  completed. 

Having  the  projections  of  the  outlines  of  the  cylinder,  the 


PEOJECTION8.  119 

projections  of  any  element  can  be  readily  obtained.  Let 
us  take,  for  example,  the  element  which  is  horizon  tall} 
projected  in  m — n.  The  vertical  projections  of  the  points 
m,  and  n,  will  be  in  the  curves  of  the  ellipses ;  the  former 
either  at  m  ,  or  m"  /  the  latter  at  71',  or  n"  /  drawing,  there- 
fore, the  lines  m' — ?i',  and  m" — n",  these  lines,  which  will 
be  parallel  to  s' — t',  will  be  the  vertical  projections  of 
two  elements  to  which  the  horizontal  projection  m — n  cor- 
responds. 

Prob.  97.  (PL  XI.  Fig.  118.)  To  construct  the  projec- 
tions',  as  in  the  last  problem,  when  the  axis  is  parallel  to  the 
vertical,  but  oblique  to  the  horizontal  plane  /  the  cylinder 
being  in  front  of  the  vertical  plane. 

This  problem  differs  from  the  preceding  only  in  that  the 
position  of  the  cylinder  with  respect  to  the  planes  of  projec- 
tion is  reversed ;  the  vertical  projection,  therefore,  in  this 
case,  will  be  a  rectangle,  having  its  sides  oblique  to  the 
ground  line ;  whilst,  in  horizontal  projection,  the  elements 
will  be  projected  parallel  to  the  ground  line,  and  the  circles 
of  the  ends  into  two  equal  ellipses. 

To  make  the  projections,  let  us  imagine  the  cylinder  rest- 
ing on  a  solid,  of  which  the  rectangle  XT  is  the  horizontal, 
and  the  one  Z  the  vertical  projection,  and  touching  the  hori- 
zontal plane  at  the  point  of  its  lower  end,  of  which  c  and  c' 
are  the  projections.  In  this  position,  the  rectangle  c'd'b'a', 
constructed  equal  to  the  generating  rectangle,  will  be  the  ver- 
tical projection  of  the  cylinder ;  the  lines  o' — -p',  and  o — -p, 
the  projections  of  the  axis ;  and  the  two  ellipses  arbt,  and 
csdu,  of  which  r — t,  and  s — u,  equal  to  the  diameters  of  the 
circles  of  the  ends,  are  the  transverse,  and  a — b,  and  c — d, 
the  horizontal  projections  of  the  diameters  parallel  to  the  ver- 
tical plane,  are  the  conjugate  axes,  the  projections  of  the  ends. 
Any  line,  as  m' — n',  drawn  parallel  to  o'—p',  will  be  the  ver- 
tical projection  of  two  elements,  the  horizontal  projections  of 
which  will  be  the  two  lines  m — n,  drawn  at  equal  distances 
from  o—p,  the  points  m,  m,  and  n,  n,  on  the  ellipses,  being 
the  horizontal  projections  of  the  points  of  which  m',  and  n' 
are  the  vertical  projections. 

Prob.  98.    (PI.  XL  Fig.  119.)     To  construct  the  projeo- 


120  INDUSTEIAL   DBAWING. 

tions,  as  in  the  list  problem ,  when  the  axis  of  the  cylinder  is 
oblique  to  both  planes  of  projection. 

To  simplify  this  case,  let  us  imagine  the  cylinder,  with  the 
solid  -Xl^,  Z  on  which  it  rests,  to  be  shifted  round,  or  re- 
volved; so  as  to  bring  them  both  oblique  to  the  vertical  plane 
of  projection,  but  without  changing  their  position  with  respect 
to  the  horizontal  plane.  In  this  new  position,  since  the  cylin- 
der maintains  the  same  relative  situation  to  the  horizontal 
plane  as  at  first,  it  is  evident  that  its  horizontal  projection 
will  be  the  same  as  in  the  preceding  case ;  the  projections  of 
the  axis  and  elements  being  only  oblique  instead  of  parallel 
to  the  ground  line.  Moreover  as  all  the  points  of  the  cylin- 
der are  at  the  same  height  above  the  horizontal  plane  in  the 
new  as  in  the  former  position,  their  vertical  projections  will 
be  at  the  same  distance  above  the  ground  line  in  both  posi- 
tions; having  the  horizontal  projection  of  any  point  in  the 
new  position,  it  will  be  easy  to  find  its  vertical  projection  by 
the  usual  method,  Prob.  94.  In  the  vertical  projection  of 
the  axis  o'—p',  in  the  new  position,  the  points  o',  and  p'  will 
be  at  the  same  distance  above  the  ground  line  as  in  the  origi- 
nal one.  The  vertical  projections  of  the  circles  of  the  ends 
will  be  the  ellipses  a'r'b't',  and  c's'd'u'.  The  vertical  projec- 
tions of  the  elements,  of  which  the  two  lines  m — n  are  the 
horizontal  projections,  will  be  the  lines  mf — n',  and  m" — n"  / 
&c.,  &c. 

Remarks.  The  preceding  problem  naturally  leads  to  the 
method  of  constructing  the  projections  of  any  body,  upon 
any  vertical  plane  placed  obliquely  to  the  original  vertical 
plane  of  projection,  when  the  projections  on  this  last  plane 
and  the  horizontal  plane  are  given.  A  moment's  reflection 
will  show  that,  whether  a  body  is  placed  obliquely  to  the 
original  vertical  plane  and  in  that  position  projected  on  it,  or 
whether  the  projection  is  made  on  a  vertical  plane  oblique  to 
the  original  vertical  plane,  but  so  placed,  with  respect  to  the 
body,  that  the  latter  will  hold  the  same  position  to  the  oblique 
plane  that  it  does  to  the  original  vertical  plane,  the  result  in 
both  cases  will  be  the  same.  If,  for  example,  G' — L'  be 
taken  as  the  trace  of  a  vertical  plane  oblique  to  the  original 
vertical  plane  of  projection,  and  such  that  it  makes  the  same 


PROJECTIONS.  121 

angle,  G'Md,  with  the  horizontal  projection  of  the  axis  of 
the  cylinder,  as  the  latter,  in  its  oblique  position  to  the  origi- 
nal ground  line  G — Z,  makes  with  this  line ;  and  the  cylin- 
der be  thus  projected  on  this  new  vertical  plane,  and  with 
reference  to  the  new  ground  line,  G' — L'  in  the  same  manner 
as  in  its  oblique  position  to  the  original  vertical  plane  of  pro- 
jection, it  is  evident  that  the  same  results  will  be  obtained  in 
both  cases ;  since  the  position  of  the  cylinder,  with  respect  to 
the  original  vertical  plane,  is  precisely  the  same  as  that  of 
the  new  vertical  plane  to  the  cylinder  in  its  original  position. 
If,  therefore,  perpendiculars  be  drawn,  from  the  horizontal 
projections  of  the  points,  to  the  new  ground  line  G' — Z',  and 
distances  be  set  off  upon  them,  above  this  line,  equal  to  the 
distances  of  the  vertical  projections  of  the  same  points  above 
G — Z,  the  points  thus  obtained  will  be  the  new  vertical  pro- 
jections. 

By  a  similar  method  it  will  be  seen  that  a  vertical  projec- 
tion on  any  new  plane  can  be  obtained  when  the  horizontal 
and  vertical  projections  on  the  original  planes  of  projection 
are  known. 

Prob.  99.  (PI.  XL  Fig.  120.)  To  construct  the  projec- 
tions of  a  right  cone  with  a  circular  base,  the  cone  resting  on 
its  sides  on  the  horizontal  plane,  and  having  its  axis  parallel 
to  the  vertical  plane,  and  in  front  of  it. 

In  this  position  of  the  cone,  its  vertical  projection  will  be 
equal  to  its  generating  isosceles  triangle,  a'b'c'  /  and  the  line 
c' — o'  drawn  from  the  projection  of  the  vertex,  and  bisecting 
the  projection  of  the  base,  a' — &',  is  the  projection  of  the 
axis. 

To  construct  the  horizontal  projection,  draw  a  line  b — c, 
at  any  convenient  distance  from  the  ground  line,  for  the  posi- 
tion of  the  horizontal  projection  of  the  axis ;  and  find  on  it 
the  horizontal  projections  c,  and  o  of  the  points  corresponding 
to  those  c',  and  o'  of  the  vertical  projection  of  the  axis ; 
through  the  point  o  drawing  a  line  d — e  perpendicular  tc 
b — c,  and  setting  off  from  o  the  distances  o — d,  and  o — £, 
equal  to  the  radius  of  the  circle  of  the  cone's  base,  the  line 
d — e  will  be  the  transverse  axis  of  the  ellipse,  into  which  the 
circle  of  the  base  is  projected  on  the  horizontal  plane.  The 


122  INDUSTEIAL  DBAWING. 

points  b,  and  a  are  the  horizontal  projections  of  the  points  of 
the  base  of  which  V  and  a'  are  the  vertical  projections ;  and 
I — a  is  the  conjugate  axis  of  the  same  ellipse. 

To  find  the  horizontal  projection  of  any  element,  of  which 
c' — m',  for  example,  is  the  vertical  projection,  find  the  hori- 
zontal projections  m,  m  corresponding  to  in'  and  join  them 
with  c  ;  the  two  lines  c — m  will  be  the  horizontal  projections 
of  the  two  elements  of  which  c' — m'  is  the  vertical  projection. 

If  it  is  required  to  find  the  horizontal  projections  of  any 
circle  of  the  cone  parallel  to  the  base,  of  which  h'—f,  paral- 
lel to  a' — b',  is  the  vertical  projection,  find  the  horizontal  pro- 
jection i  of  the  point  corresponding  to  the  projection  i'  on  the 
axis;  and  through  i  draw  p — q  parallel  to  d — e;  this  will 
give  the  transverse  axis  of  the  ellipse  into  which  the  circle  is 
projected  on  the  horizontal  plane ;  the  points  f,  and  h,  cor- 
responding to  f  and  h',  are  the  horizontal  projections  of  the 
extremities  of  the  conjugate  axis^ — h. 

Prob.  100.  (PI.  XL  Fig.  121.)  To  construct  the  projec- 
tions of  the  cone  resting,  in  the  same  manner  as  in  the  last 
problem,  on  the  horizontal  plane,  and  having  its  axis  oblique 
to  the  vertical  plane  of  projection. 

In  this  position  of  the  cone  its  horizontal  projection  will  be 
the  same  as  in  the  last  problem ;  the  vertical  projections  of 
the  different  points  will  be  found  as  in  the  case  of  the  cylin- 
der having  its  axis  oblique  to  both  planes  of  projections; 
since  the  heights  of  the  different  points  above  the  horizontal 
plane  remain  as  they  were  before  the  position  of  the  cone 
was  changed  with  respect  to  the  vertical  plane. 

Prob.  101.  If  we  imagine  the  cone  so  placed,  with  respect 
to  the  vertical  plane,  that  the  horizontal  projection  of  its  axis 
is  perpendicular  to  the  ground  line,  then  its  vertical  projec- 
tion will  be  as  represented  in  (PI.  XI.  Fig.  122),  in  which 
the  element,  or  side  on  which  the  cone  rests,  will  be  projected 
in  the  point  c' ;  and  the  circle  of  the  base  into  the  ellipse 
a'b'c'd',  of  which  a' — d',  parallel  to  the  ground  line,  and 
equal  to  the  diameter  of  the  circle  of  the  base,  is  the  transverse 
axis.  The  projection  of  any  circle  parallel  to  the  base  will  be 
constructed  in  a  like  manner. 

Prob.  102.     (PI.  XII.  Fig.  123.)     To  construct  theprojeo- 


PEOJECTION8.  123 

tions  of  a  right  hollow  cylinder  with  a  circular  base,  having 
its  axis  parallel  to  the  vertical  plane,  but  oblique  to  the  hori- 
zontal plane  of  projection  /  the  cylinder  touching  the  hori- 
zontal and  being  in  front  of  the  vertical  plane. 

This  case  differs  in  no  other  respect  from  Prob.  97,  Fig. 
118,  than  in  having  for  its  base  a  circular  ring  instead  of  a 
circle.  The  vertical  projection  of  the  outer  surface  of  the 
cylinder  will  be  the  rectangle  a'b'c'd' ;  that  of  the  interioi 
surface  a"b"c"d"  /  and  that  of  the  axis  the  line  </ — -p',  mak- 
ing the  same  angle  with  the  ground  line  as  the  axis  itself  does 
with  the  horizontal  plane. 

The  horizontal  projections  of  the  exterior  and  interior  cir- 
cles of  the  upper  end  are  the  two  concentric  ellipses  arbt,  and 
a'r'b't' /  those  of  the  lower  end  the  two  ellipses  csdu,  and 
c's'd'u'  /  and  that  of  the  axis  the  line  o — -p. 

If  we  construct  the  circular  ring  of  the  upper  base  (PI. 
XII.  Fig.  124),  and  imagine  planes  of  section  so  passed 
through  the  axis  of  the  cylinder  as  to  divide  it  into  eight 
equal  parts,  these  planes  will  cut  the  ring  of  the  upper  end 
in  lines,  as  r — m,  q — -p,  &c.,  drawn  through  the  centre  o  of 
the  exterior  and  interior  circles ;  and  the  ring  of  the  lower 
end  also  in  corresponding  lines.  The  same  planes  will  cut 
from  the  exterior  surface  of  the  sides  corresponding  elements, 
which  will  be  vertically  projected  in  the  lines jp'—p",  m' — n't 
&c.,  of  the  exterior,  and  the  lines  q' — q",  r' — r",  &c.,  of  the 
interior  surface.  The  corresponding  horizontal  projections 
of  these  lines  are  m — n,  r" — r'",  &c. 

Prob.  103.  (PI.  XII.  Fig.  125.)  To  construct  the  projec- 
tions of  the  hollow  cylinder,  as  in  the  last  problem,  the  axis 
remaining  the  same  with  respect  to  the  horizontal,  but  oblique 
to  the  vertical  plane. 

This  case  is  the  same,  in  all  material  respects,  as  in  Prob. 
98.  Fig.  119.  The  horizontal  projection  will  be  the  same  as 
in  the  last  case ;  and  the  vertical  projection  also  will  be  deter- 
mined, from  the  vertical  projection  in  the  last  case,  in  the 
same  manner  as  in  Prob.  98.  Fig.  119. 

Prob.  104.  (PI.  XII.  Fig.  126.)  To  construct  tfo projec- 
tions of  a  hollow  hemisphere,  resting  on  the  horizontal  plant 
and  in  front  of  the  vertical;  the  plane  of  section  of  tht 


124:  INDUSTRIAL   DBAWING. 

fiemisphere  being  perpendicular  to  the  vertical  plane,  but 
oblique  to  the  horizontal  plane. 

Let  a'd'o'  be  the  vertical  projection  of  the  exterior  surface  of 
the  hemisphere  ;  a"d"c"  that  of  the  interior  surface ;  and  the 
point  p',  where  the  exterior  semicircle  touches  the  ground 
line,  the  vertical  projection  of  the  point  on  which  the  hemi- 
sphere rests.  Let  the  line  a' — </,  be  the  trace  of  the  plane  of 
section,  which  forms  the  top  of  the  hemisphere,  and  which  is 
perpendicular  to  the  vertical,  and  oblique  to  the  horizontal 
plane.  This  plane  of  section  cuts  from  the  hollow  sphere  a 
circular  ring,  the  breadth  of  which  is  the  same  as  the  distance 
a! — a",  between  the  two  semicircles ;  and  the  centre  of  which 
is  vertically  projected  in  o'. 

Assuming  the  horizontal  projection  of  the  centre,  o,  at  any 
convenient  distance  from  the  ground  line,  the  diameter  of  the 
exterior  circle  of  the  top  ring,  projected  vertically  in  a'o'c', 
will  be  horizontally  projected  in  the  line  aoc,  parallel  to  the 
ground  line;  and  the  diameter  vertically  projected  in  o'  will 
be  projected  in  the  line  m — n,  drawn  through  <?,  perpendicu- 
lar to  a — c  ;  and  as  this  diameter  is  parallel  to  the  horizontal 
plane,  it  will  be  projected  on  it  into  a  line  equal  to  a' — c',  the 
diameter  of  the  ring.  The  exterior  circle  of  the  ring  will 
therefore  be  projected  into  an  ellipse,  of  which  the  lines 
a — c,  and  m — n  are  the  axes.  In  like  manner,  the  interior 
circle  will  be  projected  into  an  ellipse  of  which  the  line 
a' — c',  the  horizontal  projection  of  a" — c",  is  the  conjugate, 
and  in' — n',  equal  to  diameter  of  the  interior  circle  of  the 
ring,  is  the  transverse  axis. 

The  portion  of  the  surface  of  the  hemisphere  on  the  right, 
exterior  to  the  ring,  is  projected  in  the  semicircle  nfm  /  the 
point  f  being  the  vertical  projection  corresponding  to  f  in 
horizontal  projection. 

Prob.  105.  (PI.  XII.  Fig.  127.)  To  project  the  hemi- 
sphere, as  in  the  last  case,  the  plane  of  section  retaining  the 
same  inclination  to  the  horizontal,  but  being  oblique  to  the 
vertical  plane. 

As  the  hemisphere  has  not  changed  its  position  with  respect 
to  the  horizontal  plane,  its  horizontal  projection  will  be  the 
same  as  in  the  last  case. 


125 

The  vertical  projection  will  be  found,  as  in  similar  cases 
preceding,  by  finding  the  vertical  projections  of  the  points 
corresponding  to  the  horizontal  projections  in  the  new  posi- 
tkus,  which  will  be  at  the  same  height  above  the  ground  line 
in  the  new  as  in  the  preceding  position. 

The  exterior  surface  of  the  hemisphere  will  be  projected 
on  the  vertical  plane  in  the  semicircle  g'p'ti,  of  which  o'  is 
the  centre,  and  o' — p'  the  radius. 

The  horizontal  projection  of  the  diameter,  corresponding  to 
g' — A',  is  the  line  g — A,  parallel  to  the  ground  line. 


Intersections  of  the  preceding  Surfaces. 

The  manner  of  finding  the  projections  of  the  lines  cut  from 
surfaces  by  planes  of  section,  as  well  as  the  true  dimensions 
of  these  sections,  has  already  been  shown  (Probs.  88,  &c.), 
but  in  machinery,  as  well  as  in  other  industrial  objects,  the 
curved  portions  are,  for  the  most  part,  either  some  one  of  the 
preceding  surfaces  of  revolution,  or  else  cylindrical,  or  coni- 
cal surfaces,  which  do  not  belong  to  this  class :  and  as  these 
surfaces  are  frequently  so  combined  as  to  meet,  or  intersect 
each  other,  it  is  very  important  to  know  how  to  find  the  pro- 
jections of  these  lines  of  meeting,  or  intersection ;  and,  in 
some  cases,  even  the  true  dimensions  of  these  lines.  The 
object  then  of  this  section  will  be  to  give  some  of  the  more 
usual  cases  under  this  head. 

Prob.  106.  (PI.  XL  Fig.  128.)  To  construct  the  projec- 
tions of  the  lines  of  intersection  of  two  circular  right  cylin- 
ders, the  axis  of  the  one  being  perpendicular  to  the  horizontal 
plane^  and  of  the  other  parallel  loth  to  the  vertical  and  hori- 
zontal plane. 

In  examining  the  lines  in  which  any  two  surfaces,  whether 
plane  or  curved,  meet,  it  will  be  seen,  as  in  the  cases  of 
Probs.  88,  &c.,  if  the  two  surfaces  are  intersected  by  any 
plane  of  section,  through  a  point  of  their  line  of  meeting,  that 
this  plane  will  cut  from  each  surface  a  line ;  and  that  the 
lines,  thus  cut  from  them,  will  meet  on  the  line  of  intersection 
•)f  the  surfaces,  at  the  point  through  which  the  plane  of 


126  INDUSTRIAL  DRAWING. 

section  is  passed.  From  this,  it  will  also  be  seen  that,  in 
order  to  find  the  line  of  intersection  of  two  surfaces,  they 
must  be  intersected  by  planes  in  such  a  way  as  to  cut  lines 
out  of  the  respective  surfaces  that  will  meet ;  and  the  points 
of  meeting  of  the  lines,  thus  found,  being  joined,  will  give 
the  lines  of  meeting  of  the  surfaces.  To  apply  this  to  the 
problem  now  to  be  solved,  let  us  imagine  the  surfaces  of  the 
two  cylinders  to  be  cut  by  planes  of  section  which  will  cut 
right  line  elements  from  each,  the  points  of  meeting  of  these 
elements,  in  each  plane  of  section,  will  be  points  of  the 
required  line  of  meeting  of  the  two  surfaces. 

Let  the  circle  abed  be  the  horizontal  projection,  and  the 
rectangle  a'b'l'i'  the  vertical  projection  of  the  cylinder  per- 
pendicular to  the  horizontal  plane ;  the  point  o,  and  line 
o' — -p'  being  the  projections  of  its  axis.  Let  the  rectangles 
fghe,  and  r's's"r"  be  the  projections  of  the  other  cylinder, 
the  lines  r — s  and  g' — h'  being  the  projections  of  its  axis. 

In  these  positions  of  the  two  cylinders,  the  vertical  one 
intersects  the  horizontal  one  on  its  lower  and  upper  sides ; 
and  the  two  curves  of  intersection  will  be  evidently  in  all 
respects  the  same.  Moreover,  as  these  curves  are  on  the  sur- 
face of  the  vertical  cylinder,  they  will  be  projected  on  the 
horizontal  plane  in  the  circle  abed,  in  which  the  surface  of 
the  cylinder  is  projected.  The  only  lines  then  to  be  deter- 
mined are  the  vertical  projections  of  the  curves.  To  do  this, 
according  to  the  preceding  explanation,  let  us  imagine  the 
two  cylinders  cut  by  vertical  planes  of  section,  parallel  to  the 
vertical  plane  of  projection. 

Let  the  line  t — u,  for  example,  parallel  to  the  ground  line, 
be  the  trace  of  one  of  these  planes.  This  plane  will  cut  from 
the  vertical  cylinder  two  elements,  which  will  be  horizontally 
projected  in  the  points  m  and  n,  where  this  trace  cuts  the 
circle,  and  vertically  in  the  two  lines  m' — a?',  and  n' — z'. 
The  same  plane  will  cut  from  the  horizontal  cylinder  two 
elements,  which  will  both  be  projected  horizontally  in  the 
trace  t — u,  and  vertically  in  the  two  lines  t' — u',  and  t" — u", 
each  at  the  same  distance  from  the  line  gf — h' ;  the  one  above, 
the  other  below  this  line.  To  find  the  distance,  let  a  circle 
^,'u"h"t  &c.,  be  described,  having  its  centre  r',  on  the  line 


PROJECTIONS.  127 

prolonged,  and  with  a  radius  equal  to  * — h,  that  of  the 
base  of  the  horizontal  cylinder.  This  circle  may  be  regarded 
as  the  projection  of  the  horizontal  cylinder  on  a  vertical 
plane,  taken  perpendicular  to  the  axis  of  the  cylinder.  In 
this  position  of  the  plane  and  cylinder,  the  elements  of  the 
cylinder  will  be  projected  into  the  circumference  e'u"h",  &c., 
of  the  circle ;  and  its  axis  into  the  point  r',  the  centre  of  the 
circle.  The  two  elements  cut  from  the  cylinder,  by  the  plane 
of  which  t — u  is  the  trace,  for  example,  will  be  projected  in 
the  points  u',  and  u"  ;  each  at  the  equal  distances  v' — u', 
and  v' — u"  from  the  horizontal  diameter  e' — A",  of  this  cir- 
cle. Drawing  the  lines  t' — u',  and  t" — u",  at  the  same  dis- 
tance from  g' — Ti  as  the  points  u'  and  u"  are  above  and  below 
the  horizontal  diameter  e' — h",  they  will  be  the  projections  on 
the  primitive  vertical  plane  of  the  same  elements,  of  which 
u'  and  u"  are  the  projections  on  the  vertical  plane  perpendi- 
cular to  the  axis.  The  four  points  a?',  a?',  and  x"}  x",  in  which 
the  lines  t' — u',  and  t" — u"  cut  the  lines  m' — x',  and  n' — z' 
will  be  the  projections  of  four  points  of  the  curves  in  which 
the  cylinders  intersect.  In  like  manner,  the  vertical  projec- 
tions of  as  many  points  of  the  curves,  in  which  the  cylinders 
intersect  above  and  below  the  axis  of  the  horizontal  cylinder, 
may  be  found,  as  will  be  requisite  to  enable  us  to  draw  the 
outlines  of  their  projections. 

The  portions  of  the  curves,  drawn  in  full  lines,  lie  in  front 
of  the  two  cylinders ;  the  portions  in  dotted  lines,  lie  on  the 
other  side  of  the  cylinders. 

The  highest  points  d,  and  c  of  the  lower  curve  will  lie  on 
the  elements  of  the  vertical  cylinder,  projected  in  the  points 
d,  and  c.  The  lowest  points  y,  y  of  the  same  will  lie  on  the 
lowest  element  of  the  horizontal  cylinder,  projected  in  r — s. 
The  corresponding  points  of  the  upper  curve  will  hold  a 
reverse  position  to  those  of  the  lower. 

Prob.  107.  (PI.  XIII.  Fig.  129.)  To  construct  the  pro- 
jections of  the  curves  of  intersection  of  two  right  cylinders, 
one  of  which  is  vertical,  and  the  other  inclined  to  the  hori- 
zontal plane,  the  axes  of  both  being  parallel  to  tJie  vertical 
plane. 

Let  the  trapezoid  a'Vk'o'  be  the  vertical  projection  of  the 


128  INDUSTRIAL   DRAWING. 

vertical,  and  the  rectangle  e'f'g'ti  that  of  the  inclined  cylin- 
der. The  horizontal  projection  of  the  first  will  be  the  circle 
adbc;  that  of  the  second  figure  flihnm;  the  two  ends  of  the 
second  being  projected  into  the  two  equal  ellipses  flem,  and 
gihn. 

Intersecting  these  cylinders,  as  in  Prob.  106,  by  planes 
parallel  to  the  vertical  plane ;  any  such  plane,  as  the  one  of 
which  t — u,  for  example,  is  the  trace,  will  cut,  from  the 
inclined  cylinder,  two  elements,  projected  horizontally  in 
t — w,  and  vertically  in  the  two  lines  t1 — u'  and  t" — u"}  paral- 
lel to  the  vertical  projection  rf — s'  of  the  axis  ;  it  will  also  cut 
from  the  vertical  cylinder  the  two  elements  m}  m' — m",  and 
n,  ri — n",  the  points  #,  x,  and  a?',  a?',  where  these  projections 
cross  each  other,  will  be  points  in  the  required  projection.  A 
sufficient  number  of  like  points  being  obtained,  the  curve 
traced  through  them  will  be  the  required  vertical  projection. 
As  this  curve  lies  on  the  surface  of  the  vertical  cylinder,  its 
horizontal  projection  will  be  the  circle  adbc. 

Prob.  108.  (PL  XIII.  Fig.  130.)  To  find  the  curves  of 
intersection  of  the  two  surfaces,  as  in  the  last  case,  when  the 
cylinders  are  placed  with  the  axis  of  the  inclined  cylinder 
oblique  to  the  vertical  plane. 

In  this  new  position,  the  points  of  the  surfaces  retaining 
their  original  height  above  the  horizontal  plane,  the  new  ver- 
tical projections  of  all  the  points  will  be  at  the  same  distances 
above  the  ground  line  as  in  the  preceding  case.  Having 
therefore  drawn  the  new  horizontal  projection,  which  will  be 
the  same  as  in  the  former  case,  except  the  change  of  its  posi- 
tion to  the  ground  line,  the  vertical  projections  will  be  deter- 
mined as  in  Probs.  98,  &c. 

The  circles  of  the  ends  of  the  inclined  cylinder,  being,  in 
this  case,  oblique  to  the  vertical  plane,  will  be  projected  on 
it  in  equal  ellipses ;  the  axes  of  which  curves  are  found  as  in 
Probs.  94,  &c. 

In  this  position  of  the  cylinders,  the  projections  of  the  ele- 
ments, cut  from  the  inclined  cylinder  by  the  vertical  planes, 
as  the  one  of  which  t — u  is  the  horizontal  trace,  can  be  ob- 
tained, as  in  the  preceding  cases. 

Prob.  109.     (PL  XIII.  Fig.  131.)     To  construct  the  projec- 


PROJECTIONS.  129 

tions  of  the  curves  of  intersection  of  a  right  cone  and  right 
cylinder,  the  axis  of  the  cone  being  vertical,  that  of  the  cylin 
der  horizontal  and  parallel  to  the  vertical  plane. 

Let  abed  be  the  circle  of  the  base  of  the  cone,  and  o  the 
projection  of  its  axis ;  a'o'V  the  vertical  projection  of  the 
cone,  and  o'—p'  that  of  its  axis.  The  rectangles  efgh  and 
r's's"r",  the  projections  of  the  cylinder. 

If  we  intersect  these  surfaces  by  horizontal  planes,  any 
such  plane,  as  the  one  of  which  t — u  is  the  vertical  trace, 
will  cut  from  the  cylinder  two  elements,  of  which  t — u 
is  the  vertical  projection,  and  from  the  cone  a  circle  of 
which  m — n  is  the  vertical  projection ;  the  projections  of 
the  points,  in  which  the  elements  of  the  cylinder  cut  the 
circumference  of  the  circle  on  the  cone,  will  be  points  in  the 
required  projections  of  the  curves.  The  circle  cut  from  the 
cone  will  be  the  one  zyx,  the  radius  of  which,  o — z,  is  equal 
to  q' — in.  To  find  the  corresponding  horizontal  projections 
of  the  two  elements  cut  from  the  cylinder,  first  on  the  pro- 
longation of  f — <?',  describe  a  circle  s"u'f ',  &c.  (PI.  XIII. 
Fig.  132),  with  a  radius  s"—g",  equal  to  that  of  the  end  of  the 
cylinder,  this  circle  may  be  regarded  as  the  projection  of  the 
cylinder,  on  a  vertical  plane  perpendicular  to  the  axis  of  the 
cylinder;  the  two  elements,  cut  from  the  cylinder  by  the 
plane  of  which  t — u  is  the  trace,  will  be  projected  in  the 
points  v',  arid  v",  at  equal  distances  * — vf  and  i — v"  from  the 
vertical  diameter  s"—f  of  the  circle.  These  two  elements 
will  be  projected  on  the  horizontal  plane  in  the  two  lines 
t — w,  at  the  equal  distances  i—v',  i — v"  from  the  horizontal 
projection  n — s  of  the  axis.  The  points  y,  y  and  x,  x,  where 
these  lines  cut  the  circumference  of  the  circle  o — z,  will  be 
the  projections  of  four  points  of  the  curve ;  the  points  y'  and 
x'  are  the  corresponding  vertical  projections. 

In  the  positions  here  chosen  for  the  two  surfaces,  the 
cylinder  fits,  as  it  were,  into  a  notch  cut  into  the  cone.  The 
edges  of  this  notch,  on  the  surface  of  the  cone,  are  projected 
vertically  into  the  curvilinear  quadrilateral  d'e'c"e'd'cf;  the 
points  e'  being  the  highest,  and  c"  the  lowest  of  the  upper 
edge;  those  d'  being  the  lowest  and  c'  the  highest  of  the 
lower  edge. 


130  INDUSTRIAL   DRAWING. 

Fig.  132  represents  the  projections  of  the  surfaces  on  a  side 
vertical  plane  perpendicular  to  the  axis  of  the  cylinder. 

The  projections  of  the  same  curves,  when  the  axis  of  the 
cylinder  is  placed  obliquely  to  the  vertical  plane,  are  obtained 
by  the  same  processes  as  in  the  like  cases  of  preceding 
problems. 

Remarks.  When  the  cylinders,  in  the  preceding  Probs., 
have  the  same  diameters,  and  their  axes  intersect,  the  curves 
of  intersection  of  the  surfaces  will  be  ellipses,  and  will  be 
projected  on  the  vertical  plane  into  right  lines,  when  the 
axes  are  parallel  to  this  plane.  In  like  manner,  in  the  inter- 
section of  the  cone  and  cylinder,  the  curves  of  intersection  of 
the  two  surfaces  will  be  ellipses,  when  the  axes  of  the  two 
surfaces  intersect,  and  when  the  position  of  the  cylinder  is 
such,  that  the  circle,  into  which  it  is  projected  on  the  vertical 
plane  perpendicular  to  its  axis,  is  tangent  to  the  two  elements 
that  limit  the  projection  of  the  cone  on  the  same  plane.  In 
these  positions  of  the  two  surfaces,  if  their  axes  are  parallel 
to  the  vertical  plane,  their  curves  of  intersection  also  will  be 
projected  into  right  lines. 

Prob.  110.  (PI.  XIII.  Fig.  133.)  To  construct  the  pro- 
jections of  the  curve  of  intersection  of  a  circular  cylinder 
and  hemisphere,  the  axis  of  the  cylinder  being  vertical. 

Let  the  circle  adbc  be  the  horizontal,  and  the  rectangle 
a'Vz'x'  the  vertical  projection  of  the  cylinder ;  j>,  and p' — q' 
the  projections  of  its  axis. 

Let  the  semicircles  ghi,  be  the  horizontal,  and  g'h'i'  the 
vertical  projections  of  the  front  half  of  the  hemisphere,  being 
that  portion  alone  which  the  cylinder  enters ;  o  and  o'  the 
projections  of  its  centre. 

Any  horizontal  plane  will  cut  from  the  quarter  of  the 
sphere,  thus  projected,  a  semicircle,  and  from  the  cylinder  a 
circle ;  and  if  the  plane  is  so  taken  that  the  semicircle  and 
circle,  cut  from  the  two  surfaces,  intersect,  the  point  or  points, 
in  which  -  these  two  curves  intersect,  will  be  points  in  the 
intersection  of  the  two  surfaces  Let  m — n  be  the  vertical 
trace  of  such  a  horizontal  plane ;  it  will  cut  from  the  spherical 
surface  a  semicircle  projected  vertically  in  m. — n,  and  hori- 
zontally in  the  semicircle  mxyn,  the  diameter  of  which  is 


PBOJECTIONS.  131 

equal  to  m — n.  This  semicircle  cuts  the  circle  adbc,  which 
is  the  horizontal  projection  of  the  one  cut  by  the  same  plane 
from  the  cylinder,  in  the  two  points  x  and  y,  which  are  the 
horizontal  projections  of  the  two  points  required;  their  vertical 
projections  are  at  as'  and  y',  on  the  line  in — n.  By  a  like  con- 
struction, any  required  number  of  points  may  be  found. 

The  highest  point  of  the  projection  of  the  required  curve 
will  evidently,  in  this  case,  lie  on  that  element  of  the  cylinder 
which  is  farthest  from  the  centre  of  the  hemisphere;  and 
the  lowest  point  on  the  element  nearest  to  the  same  point. 
Drawing  a  line,  from  the  projection  o  of  the  centre  of  the 
hemisphere,  through  p,  that  of  the  axis  of  the  cylinder,  the 
points  c  and  d,  where  it  cuts  the  circle  abed,  will  give  the 
projections  of  the  two  elements  in  question.  The  vertical 
projections  of  these  points  will  therefore  be  found  at  c'  and  d', 
on  the  vertical  projections  of  the  semicircles  of  which  o — c, 
and  o — d  are  the  respective  radii. 

The  curve  traced  through  the  points  c'x'd'y',  &c.,  is  the 
vertical,  and  the  circle  adbo  the  horizontal  projection  required. 

Prob.  111.  (PI.  XIII.  Fig.  134.)  To  construct  the  pro- 
jections of  the  curves  of  intersection  of  a  right  circular  cone 
and  sphere. 

The  processes  followed,  in  this  problem,  are  in  all  respects 
the  same  as  in  the  one  preceding.  Any  horizontal  plane  will 
cut  from  the  two  surfaces  circles,  and  the  points  in  which 
the  two  circles  intersect  will  be  points  of  the  required  curve. 

Let  ghik  and  g'h'i'k'  be  the  projections  of  the  sphere  ;  o  and 
o'  those  of  its  centre ;  aybc  and  a'o'b  the  projections  of  the 
cone ;  o,  and  o' — -p'  those  of  its  axis. 

Let  m! — n'  be  the  vertical  trace  of  a  horizontal  plane.  This 
plane  will  cut  from  the  sphere  a  circle  of  which  m' — n',  and 
mxny  are  the  projections ;  and  from  the  cone  one  of  which 
r' — s',  and  rxsy  are  the  projections ;  the  points  x  and  y  in 
horizontal  projection,  and  x'  and  y'  the  corresponding  ver- 
tical projections  of  the  points  where  the  two  circles  intersect, 
are  projections  of  the  points  of  the  required  curve.  In  like 
manner,  the  projections  of  any  number  of  points  may  be 
obtained,  both  in  the  lower  and  upper  curves,  in  which  the 
cone  penetrates  the  sphere. 


132  INDUSTBIAL   DRAWING. 

The  highest  and  lowest  points,  in  the  vertical  projection  of 
the  lower  curve,  will  be  on  the  elements  of  the  cone  which  are 
farthest  from  and  nearest  to  the  centre  of  the  sphere.  These 
two  elements  are  the  ones  in  the  vertical  plane  containing 
the  axis  of  the  cone  and  the  centre  of  the  sphere;  and  are, 
therefore,  projected  in  the  line  c — y,  drawn  through  the 
points  o  and  o  /  the  line  o — c  being  the  horizontal  projec- 
tion of  the  element  nearest  the  centre ;  and  o — y  that  of  the 
one  farthest  from  it.  The  plane  which  contains  these  elements 
cuts  from  the  sphere  a  circle,  the  same  as  g'h'i'k',  and  the  points 
where  the  elements  cut  this  circle  will  be  the  highest  and  lowest 
of  the  two  curves  in  question. 

To  find  the  relative  positions  of  the  elements  cut  from  the 
cone  and  of  the  circle  cut  from  the  sphere,  we  will  use  the 
method  explained  in  Prob.  95.  For  this  purpose,  let  us 
revolve  the  plane,  containing  the  lines  in  question,  and  of 
which  c — y  is  the  horizontal  trace,  around  the  vertical  line 
projected  in  o,  and  which  passes  through  the  centre  of  the 
sphere,  until  the  plane  is  brought  parallel  to  the  vertical 
plane  of  projection  ;  in  which  new  position  its  trace  c — y 
will  take  the  position  c' — y',  turning  around  the  point  o. 
In  this  new  position,  the  circle  contained  in  this  plane  will  be 
projected  in  the  original  circle  g'h'i'k' ;  the  elements  cut  from 
the  cone  into  the  lines  a" — o"  and  V — o",  and  the  points,  in 
which  the  elements  cut  the  circle,  in/'  and  d' ';  e:  and  c". 
Taking  any  one  of  these  points,  as  the  one  df,  it  will  be  hori- 
zontally projected  in  its  new  position,  in  the  points/  and, 
when  the  plane  is  brought  back  to  its  original  position,  the 
point  s  will  come  to  z' ;  and  its  vertical  projection  will  then  be 
at  d,  the  same  height  above  the  ground  line  as  the  point  d'. 
The  points  z'  and  d  are  the  projections  of  the  highest  point  of 
the  lower  curve,  in  which  the  cone  penetrates  the  sphere.  In 
like  manner,  the  point  c",  which  is  the  lowest  point  of  the 
vertical  projection  of  the  same  curve,  may  be  found ;  also  the 
points  e  and  tf,  the  vertical  projections  of  the  highest  and 
lowest  points  of  the  upper  curve. 

The  curves,  traced  through  the  horizontal  and  vertical  pro- 
jections of  the  points  thus  obtained,  will  be  the  required 
projections  of  the  curves. 


133 


Development  of  Cylindrical  and  Conical  Surfaces. 

Cylinders  and  cones,  when  laid  on  their  sides  on  a  plane 
surface,  touch  the  surface  throughout  a  right  line  element 
If,  in  this  position,  a  cylinder  be  rolled  over  upon  the  plane, 
until  the  element,  along  which  it  touched  the  plane  in  the 
first  position  is  again  brought  in  contact  with  the  plane,  it  is 
evident  that,  in  thus  rolling  over,  the  cylinder  would  mark 
out  on  the  plane  a  rectangle,  which  would  be  exactly  equal 
in  surface  to  the  convex  surface  of  the  cylinder.  The  base 
of  the  rectangle  being  exactly  equal  in  length  to  the  circum- 
ference of  the  circle  of  the  cylinder's  base,  and  its  altitude 
equal  to  the  height  of  the  cylinder,  or  the  length  of  ita 
elements. 

In  like  manner  a  cone,  laid  on  its  side  on  a  plane,  and 
having  its  vertex  confined  to  the  same  point,  if  rolled  over  on 
the  plane,  until  the  element  on  which  it  first  rested  is 
brought  again  into  contact  with  the  plane,  would  mark  out  a 
surface  on  the  plane  exactly  equal  to  its  convex  surface;  and 
this  surface  would  be  the  sector  of  a  circle,  the  arc  of  which 
would  be  described  from  the  point  where  the  vertex  rested, 
with  a  radius  equal  to  the  element  of  the  cone  ;  the  length  of 
the  arc  of  the  sector  being  equal  to  the  circumference  of  the 
base  of  the  cone  ;  and  the  two  sides  of  the  sector  being  the 
same  as  the  element  in  its  first  and  last  positions. 

Now,  any  points,  or  lines,  that  may  have  been  traced  on 
these  surfaces  in  their  primitive  state,  can  be  found  on  their 
developments,  and  be  so  traced,  that,  if  the  surfaces  were 
restored  to  their  original  state,  from  the  developments,  these 
lines  would  occupy  upon  them  the  same  position  as  at  first. 

The  developments  of  cylinders  and  cones  are  chiefly  used 
in  practical  applications,  to  mark  out  upon  objects,  having 
cylindrical  or  conical  surfaces,  lines  which  have  been  ob- 
tained from  drawings  representing  the  developed  surfaces  of 
the  object. 

Prob.  112.  (PI.  XI.  Fig.  135.)  To  develop  the  surface 
of  a  right  cylinder  ;  and  to  obtain,  on  the  developed  surface, 
the  curved  line  cut  from  the  cylindrical  surface  by  a  plane 
oblique  to  the  axis  of  the  cylinder. 


134  INDUSTRIAL  DRAWING. 

Turning  to  Prob.  88,  Fig.  Ill,  which  is  the  same  as  the 
one  of  which  the  development  is  here  required.  Draw  a  right 
line,  on  which  set  off  a  length  a — a,  equal  to  the  circumfer- 
ence abed  of  the  cylinder's  base  ;  through  the  points  a,  con- 
struct perpendiculars  to  a — a,  on  which  set  off  the  distances 
a — a',  equal  to  the  altitude  a/ — of  of  the  cylinder ;  join 
a,' — a',  the  rectangle  aaa'a'  is  the  entire  developed  convex 
surface  of  the  cylinder.  Commencing  at  the  point  <z,  on  the 
left,  next  set  off  the  distances  a — 0,  e — c,  c — &,  &c.,  respec- 
tively equal  to  the  corresponding  portions  of  the  circumfer- 
ence a — 6,  &c. ;  and,  at  these  points,  construct  perpendicu- 
lars to  a — a.  To  find  the  developed  position  of  the  curve 
projected  vertically  in  the  points  mspn,  &c. ;  on  the  perpen- 
diculars at  a,  set  off  the  two  distances  a — m,  equal  each  to 
«' — m,  on  the  projection  of  the  cylinder;  at  e  and/J  the  dis- 
tances e — s,  andt/^ — 5,  each  equal  to  e' — s  on  the  projection  of 
the  cylinder,  &c. ;  and  so  on  for  the  other  points.  Through 
the  points  mspn,  &c.,  on  the  development,  trace  a  curve ; 
this  will  be  the  developed  curve  required. 

To  show  the  practical  application  of  this  problem,  let  us 
suppose  we  have  a  cylinder  of  any  solid  material,  on  the 
surface  of  which  we  wish  to  mark  out  the  line  that  an  oblique 
plane  would  cut  from  it.  We  would  first  make  a  drawing 
of  the  intersection  of  the  cylinder  and  plane,  as  in  Prob.  92, 
either  on  the  same  scale  as  the  given  solid,  or  on  a  propor- 
tional scale  ;  and  then  the  development  of  the  curve.  If  the 
drawing  is  on  the  same  scale  as  the  model,  the  development 
may  be  drawn  at  once  on  thick  paper,  or  thin  pasteboard ; 
and  the  paper  be  very  accurately  cut  off  along  the  developed 
curve.  Then  wrapping  this  portion  around  the  solid,  so  aa 
to  bring  the  line  a — a  to  coincide  with  its  base,  and  the 
two  edges  a — a'  to  meet  accurately,  the  curve  may  be  accu- 
rately traced  by  moving  any  sharp  pointed  instrument  care- 
fully along  the  upper  curved  edge  of  the  paper. 

Prob.  113.  (PI.  XL,  Fig.  136.)  To  develop  the  sur- 
face of  a  right  cone,  and  that  of  the  curve  cut  from  the  sur- 
face by  a  plane  oblique  to  the  axis  of  the  cone. 

Turning  to  Prob.  89  (Fig.  112),  take  the  distance  o' — a',  the 
length  of  the  element,  and  from  the  point  o'  (Fig.  136) 


PROJECTIONS.  136 

describe  an  arc.  Commencing  at  any  point,  a,  of  this  arc, 
set  off  along  it  a  length,  a — a,  equal  to  the  circumference 
aebfot  the  cone's  base  ;  next  set  off  on  this  arc  the  distances 
a — e,  e — 5,  &c.,  respectively  equal  to  those  a — e,  e — 5,  &c., 
of  the  cone's  base;  and  through  them  draw  the  radii  </ — 0, 
&c.  The  sector,  thus  obtained,  will  be  the  developed  BUI 
face  of  the  cone ;  and  the  radii  on  it  the  positions  of  the 
elements  drawn,  from  the  vertex,  to  the  points  a,  0,  5, ./"of 
the  circumference  of  the  base,  which  pass  through  the  points 
m',  r',  n'}  the  vertical  projections  of  the  curve.  The  distances 
o' — m'  and  o' — s',  from  the  vertex  to  the  points  projected  in  m' 
and  s',  are  projected  in  their  true  lengths;  setting  these  off, 
therefore,  on  the  radii  o' — a  and  o' — b  from  o'  to  m!  and  s', 
we  obtain  three  points  of  the  developed  curve.  To  obtain 
the  true  length  of  the  portion  of  the  elements  drawn  from 
the  vertex  to  the  points  projected  in  r' ;  draw,  through  r't 
the  line  x — y  parallel  to  a' — J'y  then  o' — x  will  be  this  re- 
quired length.  Setting  this  length  off,  along  the  radii  drawn 
through  c  and  d,  from  o'  to  r',  we  obtain  the  points  correspond- 
ing to  those  of  which  r'  is  the  projection.  The  curve  m'n'r's', 
&c.,  traced  through  these  points,  will  be  the  developed  curve 
required. 


136  INDUSTRIAL   DRAWING. 


CHAPTER  YIII. 

SHADOWS. 

ID  finding  the  shadows  of  objects,  the  rays  of  light  are  as- 
sumed to  be  parallel  straight  lines ;  although  the  direction  of 
light  may  be  taken  in  any  position,  it  is  customary,  as  pre- 
viously stated,  to  assume  such  a  direction  that  the  projections 
shall  make  angles  of  45°  with  the  ground  line. 

The  shadow  of  a  point  is  where  a  ray  of  Light  through  the 
point  pierces  the  surface  receiving  the  shadoio. 

To  find  the  shadow  of  a  point  upon  either  plane  of  projec- 
tion, pass  through  the  point  a  ray  of  light ;  one  projection  of 
the  ray  will  generally  meet  the  ground  line  before  the  other ; 
through  this  point  of  intersection  erect  a  perpendicular  to  the 
ground  line,  and  where  this  perpendicular  meets  the  other 
projection  of  the  ray,  will  be  the  shadow  of  the  point. 

If  the  horizontal  projection  of  the  ray  meets  GL  first,  then 
the  shadow  is  upon  F,  and  upon  H  if  the  vertical  projection 
meets  GL  first ;  if  both  projections  meet  GL  at  the  same 
point  that  will  be  the  shadow. 

To  illustrate,  let  it  be  required  to  find  the  shadow  of  the 
point  A  (Fig.  1.  PL  XXI.).  Passing  the  ray  of  light  AB 
through  the  point,  we  find  that  the  vertical  projection  of  the 
ray  meets  the  ground  line  first,  showing  that  the  shadow  is 
upon  Hj  drawing  a  perpendicular  at  V  we  get  b  as  the 
shadow.  If  a  had  been  nearer  the  ground  line  than  a'  the 
shadow  would  have  been  upon  V. 

To  obtain  the  shadow  of  a  right  line,  find  the  shadows  of 
any  two  points,  and  join. 

To  obtain  the  shadow  of  a  curved  line,  join  the  shadows  of 
a  number  of  its  points. 

The  shadow  of  a  solid  is  obtained  by  finding  the  shadows 
of  those  edges,  or  lines  of  the  surface,  which  cast  shadows. 


SHADOWS.  137 

Prob.  1.  (PL  XXI.  Fig.  2.)  To  find  the  shadow  of  a  (yule. 
The  edges  which  cast  shadows  are  a — a'b',  ae — a',fe — cV, 
f- — c'd'.  The  shadow  of  the  first  must  commence  at  a,  where 
the  line  pierces  H /  joining  this  point  with  o,  the  shadow  of 
A  (a,  a'),  we  have  the  shadow  of  the  line  a — a'b' '/  no  is  the 
shadow  of  ae — a' ;  mn  of  fa — c/af;  and/fo  of  f— c'd'. 

Notice  that  on  is  equal  and  parallel  to  ae — a',  and  am  "to 
fe — c'a' ;  whence  we  derive  the  principle  that, 

The  shadow  of  a  right  line  upon  a  plane  is  equal  and  par- 
allel to  the  line,  when  the  line  is  parallel  to  the  plane. 

Notice  also  that  ao  and/m  both  correspond  with  the  direc- 
tion of  the  projection  of  light  upon  H  j  whence  the  principle 
that, 

The  shadow  of  aright  line  upon  a  plane,  to  which  it  is  per- 
pendicular, has  the  same  direction  as  the  projection  of  light 
upon  that  plane. 

Remark.  In  drawing  these  problems  the  shadows  might 
be  made  with  parallel  lines,  or  with  a  flat  tint  of  India  ink. 

Prob.  2  (PI.  XXI.  Fig.  3.)  To  find  the  shadow  of  the 
frustum  of  a  square  pyramid.  Find,  first,  the  shadow  of  the 
vertex.  VJB,  and  V/S  are  the  edges  of  the  pyramid  which  cast 
shadows,  and  bx  and  sx  are  the  shadows  of  those  edges.  To 
find  the  shadow  of  the  point  E  it  is  sufficient  to  draw  the  hori- 
zontal projection  of  the  ray  through  e  until  it  meets  bx  at  t ; 
or  it  might  be  found  in  the  regular  way  ;  dt  is  the  shadow  of 
he — h'e' ;  the  shadow  of  ho — h'o'  is  a  parallel  through  d,  the 
intersection  of  this  with  sx  completes  the  required  shadow. 

Prob.  3.  (PI.  XXI.  Fig.  4.)  To  find  the  shadow  of  a  short 
hexagonal  prism.  The  edges  which  cast  shadows  are  c — cV, 
oc—b'c',  bd—b'c',  de—c'f,  e—f'h',  ae—h'k',  af—h'k',  and 
cf—h'o' ;  these  are  all  right  lines,  and  the  construction  of 
their  shadows  is  evident  from  the  figure. 

Find  the  shadow  when  the  prism  rests  with  one  of  its  faces 
in  the  horizontal  plane,  and  the  bases  parallel  to  V. 

Prob.  4.  (PI.  XXI.  Fig.  5.)  To  find  the  shadow  of  a  ver- 
tical cylinder.  To  determine  the  lines  of  the  convex  surface 
which  cast  shadows,  pass  vertical  planes  of  rays  tangent  to  the 
cylinder ;  en  and  tl  are  their  traces  ;  these  are  tangent  to  the 
cylinder  along  the  elements  e — e'o',  t — t'r',  the  lines  of  the 


138  INDUSTRIAL  DRAWING. 

surface  which  cast  shadows  ;  these  are  called  elements,  or  lines 
of  shade.  On  the  upper  base  that  part  of  the  circle  to  the 
right,  and  between  t  and  e,  casts  a  shadow. 

The  elements  of  shade  cast  the  shadows  enn'  and  til'. 

To  find  the  shadow  of  the  circle,  assume  points  JB,  P,  W, 
s^d  find  their  shadows  uf,  m't  x'. 

Assume  a  short  cylinder  in  a  position  similar  to  that  of  the 
prism  in  Fig.  4,  PI.  XXI.,  and  find  its  shadow  on  V. 

Prob.  5.  (PL  XXI.  Fig.  9.)  Shadow  of  a  cylindrical  aba 
cus  upon  an  octagonal  prism.  That  part  of  the  lower  base 
of  the  abacus  nearest  the  light  casts  the  shadow ;  to  determine 
how  much  of  it  casts  the  shadow,  pass  vertical  planes  of  rays 
tangent  to  the  prism,  az  and  wx  are  their  traces ;  these  planes 
intersect  the  lower  base  in  the  points  a  and  w  ;  that  part  of 
the  circle  between  these  points  will  cast  the  shadow. 

To  find  the  shadow,  assume  any  vertical  plane  of  rays  whose 
horizontal  trace  is  st ;  this  plane  intersects  the  prism  in  the 
.vertical  line  t — n't' ;  it  also  intersects  the  circle  in  the  point 
8;  the  shadow  of  8  will  be  at  the  intersection  of  the  ray 
through  8  and  the  line  t — n't',  which  gives  us  the  point  n'. 
Tn  the  same  way  other  points  of  the  shadow  may  be  found  ; 
to  find  the  points  where  the  shadow  crosses  the  edges  of  the 
prism,  proceed  in  the  same  way,  passing  the  planes  of  rays 
through  the  edges. 

Find  the  shadow  of  a  square  or  hexagonal  abacus,  upon  a 
vertical  cylinder. 

Prob.  6.  (PI.  XXI.  Fig.  6.)  Shadow  on  interior  of  hollow 
xemi-cyliiider.  Suppose  the  cylinder  to  be  vertical  and  cov- 
ered by  the  board  abed,  a' — b'  /  the  left-hand  edge  of  the 
cylinder,  and  the  lower  front  edge  of  the  board  are  the  lines 
which  cast  shadows  on  the  interior. 

The  shadow  of  the  edge  of  the  cylinder  is  found  by  pass 
ing  a  plane  of  rays  through  it ;  this  intersects  the  cylinder  in 
the  element  A,  hf — o'  /  h'  is  the  shadow  of  E.  The  shadow  of 
the  board  is  found  by  assuming  points,  as  /,  N,  etc.,  and  .find- 
ing their  shadows  in  the  same  way  that  the  shadow  of  E  was 
found. 

Find  the  shadow  on  the  interior  when  the  board  is  re- 
moved. 


SHADOWS.  139 

Prob.  7.  (PI.  XXII.  Fig.  1.)  Shadow  on  steps.  Let  the 
steps  be  given  as  in  the  figure,  there  being  a  wall  at  one  end. 
It  is  required  to  find,  first,  the  shadow  of  the  wall  on  the 
steps,  and,  second,  the  shadow  of  the  steps  upon  .^Tand  V. 

1st.  Shadow  of  wall  on  steps.  The  edges  casting  shadows 
are  a — a'b',  and  an — a'.  As  a — a'b'  is  perpendicular,  to  the 
top  of  each  step,  its  shadow  upon  these  surfaces  will  lie  in  the 
direction  of  the  projection  of  light,  as  ao,  the  part  ap  being 
on  the  top  of  the  1st  step,  and^>0  on  that  of  the  2d. 

As  a — a'b'  is  parallel  to  the  front  face  of  each  step,  its 
shadow  upon  these  faces  will  be  parallel  to  the  line  itself ; 
therefore  p'%',  found  by  projecting  up  from  ^?,  will  be  its 
shadow  upon  the  front  of  2.  The  shadow  of  A  is  at  O  on 
the  top  of  2. 

The  shadow  of  an — a'  upon  the  front  faces  of  the  steps  will 
have  the  same  direction  as  the  projection  of  light ;  while  the 
shadow  upon  the  upper  faces  will  be  parallel  to  itself ;  c'e' 
and  e'o'  are  the  shadows  upon  the  faces  of  3  and  4 ;  do  and 
he,  found  by  projecting  down  from  c'  and  e',  are  the  shadows 
on  the  tops  of  the  same  steps. 

Remark.  In  finding  the  shadow  of  A,  there  can  be  no 
question  as  to  which  surface  it  falls  upon,  if  we  remember 
that  both  projections  of  the  shadow  must  lie  in  the  projec- 
tions of  the  same  surface. 

2d.  Shadow  of  steps  on  Hand  V.  As  this  problem  pre- 
sents no  new  principles,  let  the  student,  having  first  deter- 
mined what  edges  cast  shadows,  verify  the  shadow  given 
The  shadow  on  H  is  not  complete  in  the  figure. 

Prob.  8.  (PI.  XXI.  Fig.  10.)  Shadow  of  framing.  (Fig. 
3,  PI.  XXI Y.,  shows  an  isometrical  projection  of  the  same 
framing.)  It  is  required  to  find  the  shadow  of  the  brace 
upon  the  horizontal  timbers  and  also  upon  H. 

The  edges  of  the  brace  which  cast  shadows  are  cd — c'd'  and 
ae — a'e'.  Assume  any  point  as  M  on  cd — c'd'  and  find  its 
shadow,  &,  on  the  upper  face  of  the  horizontal  timbers ;  in 
this  case  k  is  on  the  plane  of  the  top  produced ;  dk  is  the 
shadow  of  cd — c'd'  upon  the  top  of  the  timbers ;  a  parallel 
line  through  e  is  the  shadow  of  the  other  edge. 

To  obtain  the  shadow  on  H,  produce  the  ray  through  the 


140  INDUSTRIAL   DRAWING. 

point  M  until  it  pierces  H  in  i  •  draw  a  line  through  i  paral 
lei  to  dk  and  it  will  be  the  shadow  of  cd — c'd'  upon  H.  A 
line  through  q  parallel  to  it  would  be  the  shadow  of  ae — a'e' 
upon  H ;  it  does  not  show  in  this  case.  There  is  anothei 
shadow  shown  in  the  figure,  the  construction  of  which  wiL 
be  evident. 

Construct  the  shadow  of  the  whole  framing  upon  H  and  V. 

Prob.  9.  (PI.  XXI.  Fig.  8.)  Shadow  of  timber  resting 
upon  the  top  of  a  waU.  It  is  required  to  find  the  shadow  of 
the  timber  upon  the  wall,  and  also  the  shadow  of  both  upon  Y. 

The  edges  of  the  timber  which  cast  shadows,  whether  upon 
the  wall  or  upon  Y,  are  ae — a'e',  ab — a'c',  b — b'c',  bo — b'o' , 
oe — o'h')  and  e — h'ee.  The  shadow  of  ae — a'e'  upon  the  wall 
commences  at  n'  and  v'n'  is  its  shadow ;  v'g'  is  the  shadow  of 
ab — a'e' ;  g'x',  of  b — b'c' '/  the  line  through  as'  parallel  to  v'n' , 
is  the  shadow  of  bo — b'o'  upon  this  face.  The  line  bo — b'of 
also  casts  a  shadow  on  the  top  of  the  wall  which  will  be 
parallel  to  itself  through  the  point  I. 

The  construction  of  the  shadow  on  Y,  is  evident  from  the 
figure. 

Prob.  10.  (PL  XXII.  Fig.  2.)  Shadow  of  inclined  timber 
upon  triangular  prism.  The  relative  positior  of  the  pieces  is 
given  in  the  figure,  the  timber  makes  an  angle  of  60°  with  H, 
and  is  parallel  to  V. 

Commencing  with  ad — a'df  which  casts  the  shadow  ds  upon 
H,  we  find  that  this  meets  the  prism  at  v,  which  will  there- 
fore be  one  point  of  the  shadow  upon  the  nearest  inclined 
face  of  the  prism.  To  obtain  another  point,  find  the  shadow 
of  ad — a'd"  upon  an  auxiliary  horizontal  plane  through  pq — 
p'q' '/  p'q'  is  its  vertical  trace ;  e  is  the  shadow  of  A  upon  this 
plane,  and  the  line  eo,  parallel  to  ds,  is  the  shadow  of  ad — a'd' 
upon  the  same  plane ;  this  meets  pq  in  the  point  0,  which  is 
therefore  another  point  of  the  shadow  of  ad — a'd'  upon  the 
face  of  the  prism  ;  join  o  with  v,  and  we  have  the  required 
shadow. 

The  shadow  of  the  diagonally  opposite  edge  cf—c'f  can 
be  found  in  the  same  way,  or  by  drawing  fn  parallel  to  da 
and  nz  parallel  to  vo.  Part  of  the  shadow  of  the  timber  falls 
on  H  beyond  the  prism ;  ob — a'  and  bo — a'e'  are  the  onlj 


SHADOWS.  14:1 

remaining  edges  of  the  timber  which  cast  shadows.  There 
will  be  no  difficulty  in  finding  these  as  well  as  the  shadow 
of  the  prism. 

Prob.  11.  (PI.  XXII.  Fig.  4.)  To  find  the  angle  which  a 
ray  of 'light  makes  with  either  plane  of  projection.  Let  ab — 
a'V  be  the  projections  of  a  ray  of  light.  Now  if  we  consider 
the  vertical  projecting  plane  of  the  ray  to  be  revolved  into  V, 
the  ray  would  take  the  position  a"V ;  the  distance  a' a"  being 
equal  to  ca,  the  distance  of  the  point  A  in  front  of  Y ;  the 
angle  a'b'a"  is  the  angle  which  the  ray  AB  makes  with  Y, 
and  is  equal  to  35°  16'.  As  both  projections  of  the  ray  make 
the  same  angle  with  the  ground  line,  the  ray  must  make  the 
same  angle  with  H  that  it  does  with  Y. 

Prob.  12.  (PL  XXII.  Fig.  3.)  Shadow  on  interior  of 
hollow  hemisphere.  The  shadow  is  cast  by  the  semicircle  tas, 
and  it  will  be  obtained  by  finding  where  rays  through  differ- 
ent points  of  this  semicircle  pierce  the  interior.  Suppose 
the  hemisphere  projected  upon  a  plane  xy,  whicli  is  parallel 
to  the  direction  of  light  and  perpendicular  to  Y.  To  find  a 
point  of  the  shadow,  intersect  the  hemisphere  by  a  vertical 
plane  of  rays,  av  /  this  cuts  the  semicircle  av — a'e'v'  from  the 
hemisphere.  Since  the  plane  xy  is  parallel  to  the  direction 
of  light  and  perpendicular  to  Y,  rays  of  light  will  be  pro- 
jected upon  it  making  the  same  angle  with  xy  that  rays  of 
light  make  with  Y ;  that  is,  at  an  angle  of  35°  16'.  Draw  a 
ray  through  a'  at  such  an  angle,  and  project  e',  its  intersection 
with  a'e'v',  to  e  /  this  will  be  one  point  of  the  shadow  on 
the  interior. 

Other  points  may  be  found  in  the  same  way ;  the  point  o 
is  found  by  using  the  plane  bw.  The  shadow,  commences  at 
the  points  t  and  s,  where  the  projection  of  a  ray  would  be 
tangent  to  the  circle  asvt. 

Remark.  The  rays  a'e,  b'o',  &c.,  are  parallel  to  a"V  in 
Fig.  4. 

Line  of  shade.  This  is  the  line  that  separates  the  light 
from  the  dark  part  of  the  surface.  It  13  the  line  also  that 
casts  the  shadow.  This  line  can  often  be  determined  by  mere 
inspection,  as  in  the  preceding  problems,  but  in  some  cases 
special  methods  must  be  resorted  to  in  order  to  determine  it. 


142  INDUSTRIAL  DBAWINO., 

The  method  of  finding  the  line  of  shade  upon  a  vertical 
cylinder  has  been  given  in  Prob.  4,  the  elements  e — e'o'  and 
t — t'f'  being  the  lines  of  shade. 

Prob.  13.  (PI.  XXI.,  Fig.  7.)  To  find  the  line  of  shad* 
upon  a  cone.  If  a  plane  of  rays  be  passed  tangent  to  a  cone, 
the  element  along  which  it  is  tangent,  will  be  the  line  of 
shade.  As  every  plane  tangent  to  a  cone  must  contain  the 
vertex,  a  tangent  plane  of  rays  must  contain  the  ray  through 
the  vertex,  and  the  shadow  of  the  vertex  will  be  a  point  of 
its  trace ;  through  c,  the  shadow  of  the  vertex,  draw  the  line 
ce  tangent  to  the  base  of  the  cone  ;  this  is  the  trace  of  the 
tangent  plane,  and  ve — v'e'  is  the  element  of  contact  with 
the  cone ;  cd  is  the  trace  of  another  plane  tangent  on  the 
other  side  of  the  cone ;  dv — d'v'  is  its  element  of  contact. 
The  lines  ce  and  cd  are  the  shadows  of  ve — v'e',  and  dv — d'v' ', 
and  the  space  included  between  them  is  the  shadow  of  the 
cone. 

Prol,  14.  (PI.  XXII.  Fig.  5.)  To  find  the  line  of  shade 
upon  a  sphere.  Only  the  vertical  projection  of  the  sphere  is 
used,  and  the  vertical  plane  is  supposed  to  pass  through  the 
centre  of  the  sphere.  The  line  of  shade  is  the  circle  of  con- 
tact of  a  tangent  cylinder  of  rays. 

To  find  points  of  this  curve,  assume  planes  of  rays,  perpen- 
dicular to  V,  whose  traces  are  dw,  zv,  &c. ;  each  of  these 
planes  will  intersect  the  sphere  in  a  circle,  and  the  point  at 
which  a  ray  is  tangent  to  this  circle  will  be  one  point  of  the 
curve  of  shade.  The  plane  zv  intersects  the  sphere  in  a  cir- 
cle projected  in  sv  /  revolve  this  circle  about  zv  until  it  coin- 
cides with  Y;  it  will  then  have  the  position  vo'z.  To  get 
the  point  at  which  a  ray  will  be  tangent  to  this  circle,  it  is 
necessary  to  find  the  position  of  a  ray  when  revolved  in  a 
similar  manner  to  the  circle.  According  to  Prob.  11,  the  re- 
volved position  of  a  ray  would  make  an  angle  of  35°16/  with 
dw.  Now,  if  a  line  be  drawn  at  this  angle  and  tangent  to 
the  circle  vo'z,  it  will  give  o'  as  the  point  of  tangency  ;  when 
the  circle  is  revolved  back  to  its  original  position,  o'  is  pro- 
jected at  o,  and  is  one  point  of  the  curve  of  shade. 

Other  points  of  the  curve  of  shade  can  be  determined  in 
the  same  way.  The  curve  commences  at  the  points  a  and  5, 


SHADOWS.  143 

where  planes  of  rays,  parallel  to  dw,  would  be  tangent  to  the 
sphere. 

To  find  the  point  of  the  sphere  which  appears  the  bright- 
est, revolve  the  circle  dw  into  the  position  dbw  /  fc  is  the  re- 
volved position  of  a  ray  passing  through  the  centre ;  bisect 
the  angle  fob  at  m'j  when  the  circle  is  revolved  back  m'  falls 
at  m,  the  lightest  point 


INDUSTRIAL   DRAWING. 


CHAPTER  IX. 

SHADING. 

1.  Having  given  previously  the  methods  for  laying  flat 
and  graduated  tints,  let  us  see  how,  by  their  use,  we  may 
bring  out  the  true  form  of  an  object.     The  following  rules 
should  be  carefully  studied  and  followed  : 

I.  Mat  tints  should  be  given  to  plane  surfaces,  when  in 
the  light)  and  parallel  to  the  vertical  plane  /  those 
nearest  the  eye  being  lightest. 

II.  flat  tints  should  be  given  to  plane  surfaces,  when  in 
the  shade,  and  parallel  to  the  vertical  plane  ;  those 
nearest  the  eye  being  darkest. 

III.  Graduated  tints  should  be  given  to  plane  surfaces, 

when  in  the  light  and  inclined  to  the  vertical  plane  / 
increasing  the  shade  as  the  surfaces  recede  from 
the  eye  ;  when  two  such  surfaces  incline  unequally 
the  one  on  which  the  light  falls  most  directly  should 
be  lightest. 

IV.  Graduated  tints  should  be  given  to  plane  surfaces, 

when  in  the  shade,  and  inclined  to  the  vertical 
plane  ;  decreasing  the  shade  as  the  surfaces  recede 
from  the  eye. 

2.  Applying  these  rules  to  the  shading  of  an  hexagonal 
prism  (PI.  I**.  Fig.  11),  we  find  by  L,  that  the  front  face 
should  have  a  flat  tint ;  by  III.,  that  the  left-hand  face  should 
have  a  graduated  tint,  darkest  at  the  left-hand  edge  ;  by  IV., 
the  right-hand  face  has  also  a  graduated  tint,  darkest  at  the 
left-hand  edge. 

As  the  left-hand  face  receives  the  light  more  directly  than 
the  front  face,  the  nearest  part  of  it  should  be  lighter  than  the 
front  of  the  prism.  The  darkest  part  of  the  left-hand  face 
should  have  about  the  same  shade  as  the  lightest  part  of  the 
right-hand  face. 


SHADING.  14.5 

3.  The  preceding  rules  also  apply  to  the  shading  of  curved 
surfaces,  as  the  cylinder  (PI.  I**.  Fig.  12).     The  element  of 
shade  a' — V  separates  the  light  from  the  dark  part  of  the 
cylinder. 

The  surface  a — n  by  IV.  should  be  darkest  at  a  and  grow 
lighter  as  it  approaches  n. 

The  part  a — -p  of  the  illuminated  surface,  by  III.  should 
grow  lighter  as  it  approaches^?  •  the  part  m—p  by  III.  should 
be  darkest  at  ra  and  lightest  at^>/  but  by  the  second  part  of 
III.  that  part  of  the  surface  which  receives  the  light  most 
directly  should  be  the  lightest,  which  would  make  c  the 
brightest  point ;  if  then  we  take  a  point,  e,  half  way  between  c 
and  p,  it  will  give,  approximately,  the  point  which  will  appear 
the  brightest. 

The  surface  c — e  19  brighter  than  e — p  as  it  receives  the 
light  more  directly  ;  it  is  also  lighter  than  a  corresponding 
space  to  the  left  of  c,  as  it  is  nearer  the  eye ;  so  that  it  is  the 
lightest  part  of  the  cylinder. 

4.  Upon  Figs.   13  and  14,  PI.  I**,  are  shown  in  dotted 
lines  the  positions  of  the  darkest  and  lightest  parts  of  the  cone 
and  sphere.     The  method  of  finding  the  lines  of  shade  upon 
each  has  been  given  in  the  chapter  on  shadows.     As  it  is  not 
necessary  in  practice  to  locate  these  lines  exactly,  the  eye 
being  a  sufficient  guide,  it  will  be  well  to  notice  that  the  dark 
line  va  of  the  cone  is  a  little  nearer  the  right-hand  edge  than 
the  dark  line  of  the  cylinder  ;  while  the  lightest  part,  between 
vc  and  vb,  has  the  same  position  as  that  of  the  cylinder,  and 
is  determined  in  the  same  way. 

On  the  sphere,  the  point  n  of  the  line  of  shade  snp,  is  a  little 
nearer  a  than  the  centre  of  the  sphere ;  the  line  of  shade  is 
symmetrical  respecting  the  line  ba,  the  direction  of  light. 
The  lightest  point  m  is  a  little  nearer  the  centre  than  it  is  to 
b  /  it  is  also  on  the  line  ba. 

In  shading  these  solids,  commence  at  the  dark  line  and 
shade  both  ways,  using  lighter  tints  for  the  lighter  shades. 
The  dark  line  of  the  sphere  should  be  widest  at  n  and  taper 
both  ways  to  p  and  8  ;  on  the  cone  it  tapers  from  the  base  to 
the  vertex. 

10 


146  INDUSTRIAL   DRAWIKO.. 


CHAPTER  X. 

ISOMETEIOAL  DRAWING. 

If  we  take  a  cube  situated  as  in  Fig.  6,  PI.  XXII.,  and  tip 

it  up  to  the  left  about  the  point  a,e',  until  it  takes  the  position 
shown  in  Fig.  7,  the  diagonal  a'h'  being  horizontal,  and  then 
turn  the  cube  horizontally,  without  changing  its  position  with 
respect  to  H,  until  it  takes  the  position  shown  in  Fig.  8,  we 
shall  have  in  the  vertical  projection  of  Fig.  8,  what  is  called 
an  isometrical  projection. 

In  the  case  of  the  cube  it  is  the  projection  made  upon  a 
plane  perpendicular  to  a  diagonal  of  the  cube. 

The  relative  position  of  the  eye,  the  cube,  and  the  vertical 
plane  is  shown  in  Fig.  7,  where  /,  upon  a'h'  produced,  repre- 
sents the  eye  (at  an  infinite  distance) ;  xy  is  the  position  of 
the  vertical  plane  ;  the  cube  is  placed,  as  shown  by  the  pro- 
jections, so  that  the  diagonal  ah — a'h'  is  parallel  to  H  and 
perpendicular  to  xy  or  Y. 

Looking  at  this  isometrical  projection  of  the  cube  we  see 
that  the  three  visible  faces  of  the  cube  appear  equal,  and  that 
all  the  sides  of  these  faces  are  equal ;  this  shows  that  these 
faces  are  similarly  situated  respecting  V,  and  that  their  sides, 
or  the  edges  of  the  cube,  are  equally  inclined  to  Y.  It  will 
also  be  noticed  that  the  isometrical  projection  of  the  cube 
can  be  inscribed  in  a  circle,  as  the  outer  edges  form  a  regular 
hexagon. 

The  three  angles  formed  by  the  edges  meeting  at  the  cen- 
tre are  equal,  each  being  120°. 

The  point  a'  is  called  the  isometric  centre  ;  the  three  lines 
passing  through  the  centre  being  called  isometric  axes. 

Any  line  parallel  to  one  of  these  axes  is  called  an  isometric 
line,  while  any  line  not  parallel  is  called  a  non-isometric  line. 


ISOMETRICAL  DRAWING.  147 

The  "plane  of  any  two  of  the  axes,  or  any  parallel  plane,  is 
called  an  isometric  plane. 

The  two  axes  a'V  and  a'd'  (Fig.  8,  PI.  XXII.)  and  all  par- 
allel lines  make  angles  of  30°  with  a  horizontal  line. 

It  has  been  seen  that  the  isometrical  projection  of  a  cube 
can  be  inscribed  in  a  circle ;  this  renders  it  easy  to  construct 
an  isometrical  drawing  of  a  cube,  by  inscribing  a  regular 
hexagon  in  a  circle,  whose  radius  is  equal  to  an  edge  of  the 
cube,  and  then  drawing  radii  to  the  alternate  angles.  "While 
this  would  give  an  isometrical  projection  of  a  cube,  it  would 
not  be  the  true  projection  of  the  cube  whose  edge  was  taken 
as  a  radius,  because  the  edges  of  the  cube  are  inclined  to  the 
plane  of  projection,  consequently  their  projections  cannot  be 
equal  to  the  edges  themselves,  but  would  be  less. 

Let  us  see  how  the  true  isometrical  projection  of  a  cube 
may  be  obtained  without  making  it  necessary  to  construct  the 
different  projections  shown  in  Figs.  6,  7,  8,  PI.  XXII. 

The  only  lines  of  the  cube  that  are  projected  in  their  true 
size  are  the  diagonals  d'b',  d'e',  I'e'  (Fig.  8,  PI.  XXII.),  of  the 
three  visible  faces.  It  is  evident  that  the  diagonal  db — d'b' 
(Fig.  8)  -is  parallel  to  V ;  by  looking  at  Fig.  7,  where  the  rela- 
tive position  of  the  eye,  the  plane  of  projection,  and  the  cube 
is  shown,  it  will  also  be  evident  that  Ve'  (corresponding  to 
b'e',  Fig.  8)  is  parallel  to  the  plane  xy  (V). 

If  now  we  draw  a  line  db  (Fig.  9,  PI.  XXII.),  making  an 
angle  of  45°  with  db  at  the  point  b,  and  note  its  intersection 
a  with  the  vertical  through  c,  ab  will  be  the  side  of  a  square 
whose  diagonal  is  db.  and  would  therefore  be  the  true  length 
of  the  edge  of  a  cube,  the  diagonal  of  any  face  of  which  is 
equal  to  db;  but  cb  is  the  isometrical  projection  of  this  edge, 
so  that  we  have  the  means  of  comparing  the  two  and  forming 
a  scale. 

Divide  ab  into  any  number  of  equal  parts  and  project  the 
points  of  division  upon  cb,  by  lines  parallel  to  ac,  and  it  will 
give  the  isometrical  projection  of  these  distances.  To  con- 
struct then  an  isometric  scale  draw  a  horizontal  line  be  (Fig. 
10,  PI.  XXII.) ;  draw  ba  at  an  angle  of  15°  with  it ;  divide 
ba  into  any  number  of  equal  parts  and  project  the  points  of 
division  upon  be  by  lines  making  an  angle  of  45°  with  ba 


148  INDUSTRIAL   DRAWING. 

[these  projecting  lines  make  an  angle  of  60°  with  be].  The 
distances  on  be  will  be  the  isometric  length  of  the  correspond- 
ing distances  on  ba. 

This  scale  is  only  good  for  isometric  lines. 

The  diagonals  db,  dm,  bm  (Fig.  9)  are  projected  in  their 
true  lengths ;  this  would  also  be  true  of  all  lines  parallel  to 
them. 

As  ch  is  the  projection  of  a  line  equal  to  db,  a  scale  may  be 
constructed  by  projecting  any  distances,  1,  2,  3,  etc.,  from  db 
to  ch;  this  scale  will  be  good  for  all  lines  parallel  to  ch,  ho, 
or  Jin. 

Although  it  is  well  to  understand  the  construction  of  these 
scales,  they  are  seldom  used  in  practice,  as  it  is  more  conve- 
nient to  use  a  common  scale,  if  necessary,  making  the  isome- 
tric lines  equal  to  their  true  length.  This  method,  as  already 
shown,  would  make  the  drawing  larger  than  the  true  projec 
tion  of  the  object,  but  there  is  no  objection  to  this.  When 
made  in  this  way  it  is  called  an  isometrical  drawing,  to  dis- 
tinguish from  the  isometrical  projection. 

The  advantage  of  isometrical  drawing  is  that  it  offers  a 
simple  means  of  showing  in  one  drawing  several  faces  of  an 
object,  thus  obviating  the  necessity  of  a  plan  and  one  or  more 
elevations.  It  is  particularly  adapted  to  the  representation 
of  small  objects,  in  which  the  principal  lines  are  at  right 
angles  to  each  other. 

Direction  of  Light.  In  isometrical  projection  the  light  is 
supposed  to  have  the  same  direction  as  the  line  be  (Fig.  16, 
PI.  XXI1L),  the  diagonal  of  the  cube,  that  is,  it  makes  an 
angle  of  30°  with  a  horizontal  line. 

Lines  of  shade.  According  to  a  previous  definition  these 
are  the  lines  which  separate  the  light  from  the  dark  part.  In 
the  isometrical  projection  of  the  cube  (Fig.  16,  PL  XXIII.) 
the  two  right-hand  faces  (front  and  back)  and  the  bottom  are 
in  the  dark,  while  the  two  left-hand  faces  (front  and  back)  and 
the  top  are  in  the  light ;  consequently  the  heavy  lines  shown 
in  the  figure  are  the  visible  lines  of  shade. 

Prob.  1.  (Fig.  1,  PI.  XXIII.)  To  construct  the  isometri- 
cal drawing  of  a  cube,  with  a  block  upon  one  face  and  a 
recess  in  another.  Let  the  edges  of  the  cube  be  4" ;  the  block 


1SOMETKICAL   DRAWING.  149 

2"  square  and  1^-"  thick  ;  the  recess  2"  square  and  1"  deep  ; 
both  the  block  and  the  recess  to  be  ill  the  centre  of  the  face. 
The  drawing  of  the  cube  might  be  made,  as  previously  de- 
scribed, by  using  a  circle  with  a  radius  of  4",  but  a  more 
convenient  way  is  to  draw  the  isometric  axes,  ca,  cb,  and  cd, 
making  each  equal  to  4",  then  isometric  lines  through  the 
extremities  will  complete  the  cube. 

After  completing  the  cube,  divide  the  axes  into  four  equal 
parts.  To  locate  the  block,  draw  isometric  lines  through  1 
and  3,  upon  cb  and  cd,  their  intersections  will  give  the  base  of 
the  block ;  through  the  points  of  intersection  draw  isometric 
lines  parallel  to  ca,  make  them  1£"  in  length,  and  connect 
their  extremities.  The  recess  is  similarly  located,  the  depth 
of  V  can  be  obtained  by  projecting  from  2  on  either  ca  or  cb. 

Make  the  isometrical  drawing  of  a  cube  with  a  square  hole 
in  each  face,  running  through  the  cube. 

Prob.  2.  (Fig.  2,  PI.  XXIII.)  To  construct  the  isometri- 
cal drawing  of  three  pieces  of  timber  bolted  together.  Draw 
the  axes  ca,  cb,  cd;  make  cb  equal  to  6";  ce  3" ;  eo  2" ;  on 
4";  do  equals  ec  /  the  vertical  timber  is  5"x8"  ;  the  side 
timbers  are  each  let  into  the  vertical  timber  the  same  distance. 

The  method  of  constructing  the  nut  and  washer  is  given  in 
Prob.  9. 

Make  a  drawing  with  the  front  side  timber  removed. 

Prob.  3.  (Fig.  3,  PI.  XXIII.)  To  construct  the  isomet- 
rical drawing  of  a  portion  of  framing.  The  necessary 
dimensions  of  the  parts  are  given  in  Fig.  10,  PI.  XXI.  The 
edges  of  the  brace  being  non-isometric  lines,  it  is  necessary  to 
locate  the  extremities,  which  are  in  isometric  planes,  and 
then  join ;  cd  and  ca  are  each  equal  to  28" ;  the  other  edges 
are  parallel  to  ad;  de  is  equal  to  d'e'  (Fig.  10,  P.  XXI.). 

Prob.  4.  (Figs.  4,  5,  PI.  XXIII.)  To  make  the  isometri- 
cal drawing  of  a  circle.  In  Fig.  4  is  shown  a  circle  with  an 
inscribed  and  circumscribed  square.  If  we  make  an  isome- 
trical drawing  of  these  squares  in  their  relative  positions,  we 
shall  have  at  once  eight  points  through  which  the  isometric 
circle  must  pass ;  these  are  the  points  common  to  the  circle 
and  squares ;  this  is  shown  in  Fig.  5,  the  two  figures  being 
lettered  the  same.  To  locate  any  point,  as  v  (Fig.  4),  draw 


150  INDUSTRIAL   DRAWING. 

ym  perpendicular  to  ad  /  make  am  and  mv  (Fig.  5)  equal 
to  the  same  distances  on  Fig.  4.  in  this  way  any  point  of  the 
circle,  or  any  point  within  the  square,  can  be  located.  This 
method  gives  an  exact  drawing  of  the  circle,  the  curve  being 
an  ellipse. 

Prob.  5.  (Fig.  6,  PI.  XXIII.)  To  make  an  approximate 
construction  of  the  isometrical  drawing  of  a  circle.  Con- 
struct the  isometric  square  abed  ;  let  d  be  the  centre  and  dn 
the  radius  of  the  arc  nx  •  t>  is  the  centre  of  the  arc  pq,  the 
radius  being  the  same  as  before ;  s  and  o  are  the  centres  of 
the  arcs  np  and  qx  /  the  points  n,p,  #,  x,  are  the  centres  of 
the  sides. 

The  curve,  as  thus  constructed,  approximates  near  enough 
to  the  true  curve  to  answer  most  purposes. 

Make  an  isometrical  drawing  of  a  cube  with  a  circle  in- 
scribed in  each  face. 

Prob.  6.  (Fig.  7,  PI.  XXIII.)  To  divide  the  isometrical 
drawing  of  a  circle  into  equal  parts. 

1st  method.  At  n,  the  centre  of  ab,  erect  the  perpendicular 
nc',  and  make  it  equal  to  na  ;  from  c'  as  a  centre  describe  the 
arc  mp  and  divide  it  into  any  number  of  equal  parts  ;  draw 
lines  through  these  points  from  c'  and  produce  them  until 
they  meet  ab ;  join  the  points  on  ab  with  c,  and  the  lines 
drawn  will  divide  the  isometric  arc  into  the  same  number  of 
parts  that  mp  contains. 

%d  method.  Describe  the  semicircle  de'h  upon  dh  as  a  dia- 
meter ;  this  is  the  semicircle  of  which  deh  is  the  isometrical 
drawing  and  is  in  the  position  deh  would  take  when  revolved 
about  dh,  as  an  axis,  until  parallel  to  V.  Divide  de'h  into 
equal  parts  and  project  to  deh  by  vertical  lines ;  these  will 
divide  the  isometric  curve  into  a  corresponding  number  of 
parts. 

Prob.  7.  (Fig.  8,  PL  XXIII.)  To  make  the  isometrical 
drawing  of  a  cube,  cylinder,  and  sphere.  Suppose  the  sphere 
to  rest  upon  the  top  of  the  cylinder,  and  the  cylinder  upon 
the  cube. 

The  diameter  of  the  sphere  and  cylinder  is  equal  to  an 
edge  of  the  cube ;  the  height  of  the  cylinder  is  equal  to  itf 
diameter. 


ISOMETEICAL  DRAWING.  151 

First  construct  a  cube  and  then  inscribe  a  circle  in  the  top 
face ;  this  is  the  lower  base  of  the  cylinder ;  to  obtain  the 
upper  base,  construct  a  second  cube  resting  upon  the  first,  and 
of  the  same  size  ;  this  is  shown  in  the  figure  by  dotted  lines. 
Inscribe  a  circle  in  the  top  of  this  second  cube  and  it  will  be 
the  upper  base  of  the  cylinder ;  ab  and  cd  tangent  to  each  of 
these  curves  are  the  extreme  elements  of  the  cylinder.  The 
sphere  rests  upon  the  centre  of  the  upper  base ;  erect  a  per- 
pendicular at  <?,  and  make  it  equal  to  half  the  edge  of  the 
cube  ;  from  h  as  a  centre,  with  a  radius  equal  to  de,  describe 
the  sphere. 

Prob.  8.  (Figs.  9,  10, 11,  PI.  XXIII.)  Isometrical  draw- 
ing of  brackets  supporting  a  shelf.  No.  1  of  the  brackets  has 
all  of  its  edges  right  lines,  while  No.  2  is  made  up  partly  of  arcs 
of  circles,  whose  centres  are  at  h  and  m  (Fig.  11).  Figs.  10 
and  11  give  side  elevations  of  the  two  brackets,  with  dimen- 
sions. 

The  shelf  is  28"  wide  and  §"  thick ;  make  the  projection 
so  as  to  show  the  under  side. 

There  will  be  no  difficulty  in  making  the  drawing  of  No.  1. 

To  construct  No.  2,  after  drawing  bo,  ca,  ae,  ed,  dg,  ew, 
locate  A,  this  is  the  centre  of  the  isometric  squares  which 
contain  the  curves  en  and  ou  ;  therefore,  draw  the  isometric 
line  hs  and  make  it  equal  to  ho  or  he  ;  s  is  the  centre  of  the 
curve  en  ;  st  is  equal  to  the  thickness  of  the  bracket,  and  t  is 
the  centre  of  the  curve  through  w.  Make  ht  equal  to  hv  and 
t  is  the  centre  of  the  curve  ou  ;  p  is  the  centre  of  the  curve 
In,  and  tz  the  centre  of  gu  and  also  of  the  curve  through  x. 

Prob.  9.  (Fig.  12,  PL  XXIII.)  The  isometrical  drawing 
of  a  nut  and  washer.  The  washer  is  3£"  in  diameter  and  £" 
thick;  the  nut  2"  square  and  1"  thick;  the  bolt  1"  in 
diameter. 

To  construct  the  washer,  make  two  isometric  squares,  ab, 
cd,  %"  apart  and  3£"  on  a  side ;  inscribe  in  each  of  these  a 
circle  and  connect  by  isometric  lines  at  e  and  h,  tangent  to 
each  circle. 

Find  the  centre  of  the  washer  and  construct  the  square  for 
the  base  of  the  nut. 

After  completing  the  nut,  construct  in  the  centre  of  its 


152  INDUSTRIAL   DRAWING. 

face  an  isometric  square  V  on  a  side  ;  inscribe  in  this  a  circle, 
which  will  represent  the  curve  of  intersection  of  the  bolt  with 
the  nut.  In  the  figure  the  bolt  is  supposed  to  project  V  from 
the  nut ;  the  rest  of  the  construction  will  be  apparent  from 
the  figure. 

In  Fig.  12,  the  nut  is  placed  so  that  its  edges  are  isometric 
lines ;  to  make  a  drawing  when  the  nut  is  turned  so  that  its 
edges  would  not  be  isometric  lines,  make  a  plan  of  the  nut 
and  washer  as  in  Fig.  13,  PL  XXIII.,  and  draw  through  the 
corners  of  the  nut  a  square,  abed,  with  its  sides  parallel  to 
the  sides  of  the  square  which  circumscribes  the  washer.  Af- 
ter constructing  the  washer  as  before,  make  in  the  centre  of 
the  washer  an  isometrical  drawing  of  the  square,  abed,  and 
locate  on  it  the  points  ehon  /  the  method  of  completing  will 
be  the  same  as  before. 

Make  a  drawing  when  the  nut  is  oblique. 

Prob.  10.  (Figs.  14  and  15,  PI.  XXIII.)  The  isometrical 
drawing  of  letters.  In  these  letters  the  position  of  the  iso- 
metric axes  is  changed,  one  being  placed  in  a  horizontal 
position.  The  construction  of  these  letters  is  apparent  from 
the  figures ;  in  the  letter  xS,  the  curves  not  being  isometric 
circles,  will  have  to  be  sketched  in  or  drawn  with  an  irregular 
curve. 

Try  other  letters  of  the  alphabet,  as  d,  w,  x,  o. 

Shadows.  In  isometrical  drawing  the  shadow  of  a  point 
on  any  plane  surf  ace  is  at  the  intersection  of  the  ray  through 
the  point,  and  the  projection  of  the  ray  on  the  surface. 

We  have  seen  that  in  isometrical  projection  rays  of  light 
make  an  angle  of  30°  with  a  horizontal  line. 

Turning  to  Fig.  16,  PI.  XXIII.,  let  be,  the  diagonal  of  the 
cube,  represent  a  ray  of  light ;  bd  is  the  projection  of  this 
ray  on  the  top  of  the  cube,  and  bm  the  projection  on  the  face 
boms.  Thus  we  see  that  the  position  of  the  projection  of  a 
ray  of  light  upon  a  horizontal  plane,  as  bnds,  is  horizontal, 
while  the  projection  of  a  ray  of  light  upon  a  vertical  plane, 
as  boms,  makes  an  angle  of  60°  with  a  horizontal  line. 

Suppose  the  edge  bo  produced  to  a  and  it  is  required  to 
find  the  shadow  of  a  ;  draw  the  ray  ag  at  30° ;  through  b, 
the  projection  of  a  upon  the  plane  of  the  top,  draw  the  hori- 


ISOMETKICAL  DRAWING.  153 

zontal  line  Ig ;  this  is  the  projection  of  the  ray,  and  the 
intersection  of  this  with  ag  is  the  shadow  of  a  upon  the  top 
of  the  cube. 

Produce  the  edge  In  to  c;  the  shadow  of  c  is  at  h,  the  in- 
tersection of  the  ray  ch  with  bh,  its  projection  upon  the  vertical 
face. 

Prob.  11.  (Fig.  16,  PI.  XXIII.)  The  isometrical  drawing 
of  a  cube  with  its  shadow  on  the  horizontal  plane.  The 
shadow  of  s  is  at  t  /  tm  is  the  shadow  of  sm  /  w  is  the  shadow 
of  d,  and  tw  the  shadow  of  sd ;  vw  is  the  shadow  of  nd,  and 
vs  the  shadow  of  ns,  the  back  edge. 

Prob.  12.  (Fig.  2,  PI.  XXIY.)  The  isometrical  drawing 
of  an  hexagonal  prism  with  the  shadow.  The  prism  is  rep- 
resented standing  on  H  at  a  distance  xy  from  V.  The  edges 
which  cast  shadows  are  ia,  ab,  be,  cd,  de.  Since  ia  is  perpen- 
dicular to  H,  its  shadow  upon  that  plane  will  have  the  same 
direction  as  the  projection  of  light  upon  H ;  the  shadow  of  ia 
falls  partly  on  Y,  as  at  ok,  which  is  parallel  to  ia  ;  k,  the 
shadow  of  a,  is  at  the  intersection  of  the  ray  through  a, 
and  the  vertical  through  o.  In  the  same  way  I,  m,  n,  the 
shadows  of  b,  c,  d,  are  found. 

In  finding  the  shadow  of  the  prism  upon  V.  the  vertical 
projection  of  the  rays  of  light  have  not  been  used,  as  they 
were  not  necessary. 

Prob.  13.  (Fig.  3,  PI.  XXIY.)  The  isometrical  drawing 
of  a  beam,  projecting  from  a  vertical  wall,  with  the  shadow. 

The  construction  of  the  drawing  is  evident  from  the  figure. 
The  edges  which  cast  shadows  are  ca,  ab,  bd,  dn.  Only  the 
projections  of  the  rays  are  used  to  find  the  shadow.  To  find 
the  shadow  of  a,  draw  through  c,  the  vertical  projection  of  a, 
the  line  ce  at  an  angle  of  60° ;  through  h,  the  horizontal  pro- 
jection of  a,  draw  the  horizontal  line  hi;  these  two  lines  ce 
and  hi  are  the  projections  of  the  ray  through  a,  and  they 
determine  e,  the  shadow  of  a.  The  accuracy  of  this  construc- 
tion can  be  tested  by  drawing  a  ray  through  a.  In  the  same 
way  the  points  i  and  k  are  found. 

Prob.  14  (Fig.  5,  PI.  XXIY.)  The  isometrical  drawiig 
of  a  four-armed  cross,  with  the  shadows.  It  is  required  to 
find  the  shadow  of  the  cross  on  H,  and  also  upon  itself ;  a—I 


154  INDUSTRIAL   DRAWING. 

is  the  shadow  of  ae ;  1 — 2  the  shadow  of  gf ;  2—3  of  fh; 
3—4  of  li;  4—5  of  Im;  5—6  of  bo;  6—7  of  en;  7—8  of 
cq;  the  rest  of  the  shadow  is  cast  by  edges  upon  the  back, 
tvhich  are  not  seen. 

The  shadows  upon  the  cross  are  si  the  shadow  of  sh  ;  sm. 
of  bs  /  he  of  hg  ;  bq  of  bn. 

Keeping  the  same  thickness  of  the  cross  make  the  arms 
longer,  and  then  find  the  shadows. 

Prob.  15.  (Fig.  1,  PI.  XXIY.)  The  isometrical  drawing 
of  a  vertical  cylinder  passing  through  an  hexagonal  block, 
with  the  shadows.  A  plan  of  the  cylinder  and  block  is  shown 
in  Fig.  4 ;  the  diameter  of  the  cylinder  is  •$/' ;  any  edge  of 
the  block,  as  be,  is  ^"  in  length;  the  block  is  fj."  thick. 
First  make  an  isometrical  drawing  of  this  plan  ;  this  is  shown 
in  Fig.  1,  with  the  points  numbered  and  lettered  the  same. 
At  any  assumed  distance  above  the  plan  construct  a  similar 
figure  ;  this  gives  the  top  of  the  block,  which  can  readily  be 
completed.  The  method  of  completing  the  cylinder  will  be 
evident  from  what  has  preceded. 

Shadows.  Find  first  the  shadow  which  the  block  casts 
upon  the  cylinder.  Assume  any  vertical  plane  of  rays  as  the 
one  whose  horizontal  trace  is  vx ;  this  cuts  the  point  v' 
from  the  lower  edge  of  the  block,  and  the  element  xx '  from 
the  cylinder;  the  ray  through  v'  intersects  the  element  at  #' 
one  point  of  the  required  shadow.  In  the  same  way  other 
points  can  be  found ;  Jcy  is  the  element  of  shade  found  by 
passing  the  plane  of  rays  wy  tangent  to  the  cylinder.  At  z 
the  shadow  passes  to  the  back  part  of  the  cylinder. 

The  method  of  finding  the  shadow  which  the  cylinder  casts 
upon  the  block  will  be  evident  from  the  figure,  with  what  has 
just  been  shown. 

Show  how  the  shadows  given  on  Fig  3,  PI.  XXIII.,  are 
found. 

Determine  whether  the  brace  would  cast  a  shadow  on  the 
top  of  the  horizontal  timber  or  not. 

Find  the  shadow  of  the  framing  upon  H. 


OBLIQUE  PBOJEOTION.  155 


CHAPTER  XL 

OBLIQUE  PROJECTION. 

This  method  of  projection  is  similar  to  isometrical  projec- 
tion in  showing  three  faces  of  the  object,  but  unlike  isometri- 
cal it  gives  the  exact  form  of  one  of  these.  It  is  called 
oblique,  because  the  projecting  lines  are  oblique  to  Y  instead 
of  being  perpendicular,  as  they  have  been  previously. 

It  is  sometimes  called  Parallel  Perspective  ;  it  does  not  how- 
ever give  a  true  perspective  of  the  object,  but  offers  a  substi- 
tute, simple  in  construction,  and  one  as  well  adapted  for  rep- 
resenting small  objects. 

An  oblique  projection  of  a  cube  is  given  in  Fig.  11,  PI. 
XXIV. ;  the  face  abed  has  its  true  form,  while  the  other  two 
faces  are  shown  equally,  but  not  in  their  true  shape.  The 
edges  ce,  bh,  ai,  make  an  angle  of  45°  with  a  horizontal  line, 
and  are  equal  to  the  other  edges  of  the  cube. 

Since  the  face  abed  is  projected  in  its  true  form,  it  must 
be  parallel  to  V ;  if  then  we  suppose  the  cube  placed  with 
one  face  parallel  to  V,  it  is  evident  that  this  projection  (Fig. 
11)  could  not  be  obtained,  when  the  projecting  lines  are  per- 
pendicular to  Y ;  the  square,  abed,  would  be  the  projection  in 
that  case.  In  order  then  to  obtain  this  projection,  the  pro- 
jecting lines  cannot  be  perpendicular  to  Y,  but  must  be 
oblique. 

Now  when  a  line  is  perpendicular  to  a  plane,  in  order  that 
the  projection  of  the  line,  upon  that  plane,  should  be  equal  to 
the  line  itself,  the  projecting  lines  must  make  an  angle  of  45° 
with  the  plane.  In  the  case  of  a  line  parallel  to  a  plane,  its 
projection  would  be/  equal  to  the  line,  whatever  the  direction 
of  the  projecting  lines,  provided  they  were  parallel. 

Thus  we  see  that  if  the  projecting  lines  make  an  angle  of 
45°  with  V,  we  shall  obtain,  in  the  case  of  the  cube,  the  pro- 
jection shown  in  Fig.  11. 


156  INDUSTRIAL  DBA  WING. 

Since  an  infinite  number  of  projecting  lines  could  be  passed 
through  a  point,  all  making  the  required  angle  of  45°,  it  fol- 
lows that  the  projection  of  a  line  might  have  an  infinite 
number  of  positions.  Thus  the  edges  ce,  bh,  etc.  (Fig.  11), 
could  be  drawn  at  any  angle,  and  still  give  an  oblique  projec- 
tion of  the  cube.  Fig.  8,  PI.  XXIV.,  gives  an  oblique  pro- 
jection of  the  cube,  with  the  edges  ce,  bh,  etc.,  drawn  at  an 
angle  of  30° ;  other  angles  might  be  used,  but  these  two  (45° 
and  30°)  are  most  convenient,  as  they  are  found  on  the  trian- 
gles. 

Only  those  lines  that  are  parallel  or  perpendicular  to  V  are 
projected  in  their  true  size. 

Prob.  1.  (Fig.  12,  PI.  XXIV.)  To  make  the  oUique pro- 
jection of  a  circle.  Let  abed  be  the  circumscribed,  and  liken 
the  inscribed  squares  of  the  circle  m/poq.  Construct  a'b'c'd', 
and  h'k'e'n' ',  the  oblique  projection  of  these  squares,  and  we 
shall  have  eight  points  (m'n'p'e'o'k'q'h'}  of  the  oblique  pro- 
jection of  the  circle.  The  method  of  finding  other  points  will 
readily  suggest  itself,  but  the  points  already  found  will  be 
sufficient  to  enable  us  to  trace  the  curve  quite  accurately. 
The  curve  thus  found  would  be  an  ellipse ;  for  an  approxi- 
mate construction  of  it  by  arcs  of  circles,  draw  the  lines  m's 
and  q's  perpendicular  to  the  sides  of  the  square,  their  inter- 
section, s,  will  give  a  centre  for  the  curve  q'h'm' /  in  the  same 
way  the  curve  p'e'o'  can  be  drawn  as  an  arc  ;  for  the  ends  use 
i)  and  w  as  centres. 

This  figure  (12)  represents  the  projection  of  the  circle  when 
horizontal,  as  upon  the  top  of  the  cube.  "When  the  circle  is 
parallel  to  V,  or  upon  the  front  face  of  the  cube,  its  projec- 
tion would  be  an  equal  circle  ;  this  is  shown  in  Fig.  9,  PL 
XXIY.,  which  is  the  oblique  projection  of  three-fourths  of  a 
hollow  cylinder,  whose  axis  is  perpendicular  to  V,  the  ends 
being  projected  as  circles. 

Fig.  6,  PI.  XXIY.,  gives  an  example  of  the  oblique  projec- 
tion of  the  circle  when  perpendicular  to  both  ZT  and  V,  as 
upon  tlie  right-hand  face  of  the  cube. 

Construct  an  oblique  projection  of  a  cube  with  a  circle  upon 
each  of  the  three  visible  faces. 

Direction  of  light.     The  light  is  assumed,  as  in  isometricul 


OBLIQUE   PBOJECTION.  157 

projection,  to  have  the  same  direction  as  the  diagonal  of  the 
cube.  The  arrow  in  Fig.  11,  PI.  XXIV.,  indicates  the  direc- 
tion of  light,  corresponding  to  the  direction  of  the  diagonal  ae; 
ah  is  the  projection  of  this  ray  (ae)  upon  the  top  of  the  cube, 
and  ac  the  projection  upon  the  front  face.  The  method  of 
finding  shadows  is  similar  to  that  in  isometrical  projection. 
The  shadow  of  the  cube  upon  If  is  shown  in  Fig.  11 ;  if  the 
shadow  of  the  cube  (Fig.  8,  PL  XXIV.)  was  to  be  found,  the 
direction  of  light  would  correspond  with  the  direction  of  its 
diagonal  ae. 

Only  a  few  examples  of  this  method  of  projection  are  given, 
sufficient,  however,  to  show  its  application.  Fig.  7,  PL  XXIV., 
represents  an  irregular  block ;  it  is  4"  long  and  1"  square  on 
the  ends,  the  short  lines  are  %"  in  length.  Fig.  10,  PL 
XXIV.,  is  the  oblique  projection  of  a  mortise  and  tenon ;  the 
under  side  of  the  tenon  is  shown  ;  notice  the  position  of  the 
shade  lines  on  the  upper  piece. 


158  INDUSTRIAL  DRAWING. 


CHAPTER  XIL 

LINEAR  PERSPECTIVE. 

Linear  perspective  is  the  representation  of  the  fcrm  of  an 
object  upon  any  plane  surface,  just  as  it  appears  to  the  eye 
when  viewing  it  from  any  given  point.  To  use  a  common  il- 
lustration :  if  we  should  close  one  eye,  and,  keeping  the  other 
at  a  fixed  distance  from  the  window,  should  trace  upon  the 
glass  the  outlines  of  what  could  be  seen  through  it,  we  should 
have  a  true  perspective  of  the  objects  seen. 

Supposing  then,  the  perspective  plane .  to  be  transparent, 
and  always  placed  between  the  eye  and  the  object,  we  see  that 
the  perspective  of  a  point  is  where  a  visual  ray  (a  line 
drawn  from  the  point  to  the  eyo)  pierces  the  pcrspectie  plane. 
To  illustrate,  let  A  (Fig.  1,  PI.  XXV.)  be  a  point  in  space, 
behind  the  vertical  plane  which  is  used  as  the  perspective 
plane ;  C  is  the  position  of  the  eye,  or  point  of  sight ;  the 
the  visual  ray  CA  pierces  V  at  #",  which  is  the  perspective 
of  A. 

It  is  impossible  in  practice  to  draw  the  visual  ray  itself,  as 
in  Fig.  1  ;  the  point  of  sight  and  the  point  in  space  are  given 
by  their  projections  only,  so  that  it  is  necessary  to  use  the 
projections  of  the  visual  ray  to  find  the  perspective.  To  do 
this,  join  the  horizontal  projection  of  the  point  of  sight  with 
the  horizontal  projection  of  the  point,  join  also  the  vertical 
projections;  the  two  lines  thus  drawn  are  the  projections  of 
the  visual  ray,  and  the  point  in  which  it  pierces  Y  is  the  re- 
quired perspective.  In  Figs.  1  and  2,  PL  XXV.,  the  visual 
ray  ca — c'a!  pierces  V  at  a".  Fig.  2  has  the  same  letters  and 
measurements  as  Fig.  1,  and  represents  the  same  thing,  with 
the  customary  position  of  the  planes  of  projection. 

It  will  be  noticed  in  Fig.  2,  that  both  projections  of  the 
point  A  are  above  the  ground  line  ;  as  this  has  not  happened 
before,  a  word  or  two  in  explanation.  Looking  at  Fig.  1  we 


LINEAR  PEBSPECTIYE.  159 

Bee  that  the  two  intersecting  planes  of  projection  fo>  ID  four 
dihedral  angles ;  the  one  above  H  and  in  front  of  Y,  is 
called  the  1st ;  the  2d  is  behind  Y  and  above  H ;  the  3d  is 
below  the  2d,  and  the  4th  below  the  1st.  The  point  of  sight 
is  in  the  first  angle.  Now,  all  the  objects  that  we  have  pre 
viously  considered  have  been  placed  in  the  first  angle,  and,  as 
we  have  seen,  the  horizontal  projections  are  always  below  the 
ground  line,  and  the  vertical  above ;  but  if  an  object  is 
placed  in  the  second  angle  (as  it  is  in  perspective)  when  the 
vertical  plane  is  revolved  back,  both  projections  will  appear 
above  the  ground  line,  as  in  Fig.  1,  where  a'  revolves  to  a'. 
The  same  rules  that  we  have  had  for  determining  the  posi 
tion  of  a  point  in  space  from  its  projections,  hold  good  when 
both  projections  are  above  the  ground  line  ;  the  distance  ah 
(Figs.  1  and  2)  is  the  distance  of  A  from  Y,  and  the  distance 
a'h  is  the  distance  of  A  from  H. 

Prob.  1.  (Figs.  3,  4,  5,  PI.  XXY.)  To  find  the  perspec- 
tive of  right  lines  in  different  positions  by  means  of  visual 
rays. 

1st,  when  perpendicular  to  H :  let  a — a'b'  (Fig.  3)  be  a 
line  perpendicular  to  H,  and  at  the  distance  ab'  behind  Y ; 
cc'  is  the  point  of  sight.  Draw  the  ray  ac — a'c' /  this  pierces 
Y  at  0,  the  perspective  of  the  point  aa' ;  /is  the  perspective 
of  the  point  ab' ;  fe  is  the  perspective  of  the  line  a — a'b'; 
in  the  same  way  is  found  hg,  the  perspective  of  the  line 
d — a'b',  which  is  parallel  to  a — a'b'. 

2d,  when  parallel  to  H  and  V:  let  a'b— a'b'  (Fig.  4)  be 
the  given  line,  situated  in  H  and  at  the  distance  bb'  behind 
Y ;  the  perspective  of  the  point  aa!  is  at  e  ;  f  is  the  perspec- 
tive of  the  point  bb',  whence  efv&  the  perspective  of  ab — a'b'; 
hg  is  the  perspective  of  the  line  dm — a'b',  which  is  parallel 
to  ab — a'b'. 

3d,  when  perpendicular  to  V:  ab — b'  (Fig.  5)  is  the  given 
line,  situated  in  H,  and  at  the  distance  bb'  from  Y ;  ef  is  its 
perspective ;  in — ri  is  a  line  parallel  to  ab—b';  hb  is  its  per- 
spective. 

Prob.  2.  (Fig.  7,  PI.  XXY.)  To  find  the  perspective  of 
a  cube  when  placed  with  a  face  parallel  to  V.  Let  abid,  and 
a'd'fe'  be  the  projections  of  the  cube,  and  cc'  the  point  of 


160  INDUSTRIAL   DRAWING. 

Bight.  Find  first  the  perspective  of  the  point  df,  by  drawing 
the  ray  cd — c'f;  this  gives  o  as  the  perspective  ;  the  ray 
ca — c'e?  gives  r,  the  perspective  of  the  point  ae' ;  t  is  the  per- 
spective of  aa'j  and  s  of  ddf;  in  the  same  way  are  found 
the  points  j>,  q,  v,  z. 

Prob.  3.  (Fig.  6,  PI.  XXV.)  To  find  ike  perspective  of 
a  cube  when  placed  with  its  faces  oblique  to  V.  Let  abid-, 
and  a'i'h'e'  be  the  projections  of  the  cube ;  s  is  the  perspec- 
tive of  the  point  dd'j  o  of  the  point  df  ;  r  of  ae'j  t  of  aa'} 
etc. 

By  an  examination  of  Figs.  3,  4,  6,  7,  PI.  XXV.,  it  will  be 
seen  that  the  perspective  of  any  right  line  is  parallel  to  that 
line  when  the  line  is  parallel  to  the  perspective  plane. 

It  will  also  be  noticed  that  the  edges  op,  sq,  tv,  &c.  (Figs. 
6  and  7),  when  produced,  meet  at  a  point ;  this  is  called  the 
vanishing  point  of  these  lines,  and  since  the  lines  op,  sq,  tv, 
etc.,  are  the  perspectives  of  parallel  lines,  we  have  the  prin- 
ciple that  the  perspectives  of  all  parallel  lines  have  a  com- 
mon vanishing  point. 

To  find  then  the  vanishing  point  of  any  line,  draw  a  line 
through  the  point  of  sight  parallel  to  the  given  line,  and 
where  it  pierces  the  perspective  plane  will  be  the  vanishing 
point  of  this  line,  and  all  parallel  lines.  In  Fig.  6,  cw — c'w' 
is  parallel  to  di — d'i';  it  pierces  Y  at  w',  which  is  the  vanish- 
ing point  of  di — d'i',  and  the  edges  parallel  to  it.  The  van- 
ishing point  of  ro,  ts,  vq,  etc.,  could  be  found  in  the  same  way. 

The  vanishing  point  of  parallel  lines,  parallel  to  the  per- 
spective plane,  is  situated  at  an  infinite  distance  ;  hence  the 
perspectives  will  be  parallel. 

In  Figs.  5  and  7,  PI.  XXY.,  the  lines  which  are  perpen- 
dicular to  Y  vanish  at  cr ;  whence  the  principle  that  all  lines 
perpendicular  to  the  perspective  plane  vanish  in  the  vertical 
projection  of  the  point  of  sight ;  this  is  called  the  centre  of 
the  picture. 

The  horizontal  line  through  the  centre  of  the  picture  is 
called  the  horizon. 

The  vanishing  points  of 'all  horizontal  lines  are  situated 
somewhere  on  the  horizon. 


LINEAR   PERSPECTIVE.  16t 

The  point  in  which  a  line  pierces  a  plane  is  called  ita 
trace ;  the  trace  of  a  line  en  the  perspective  plane  is  one 
point  of  its  perspective ;  the  vanishing  point  of  the  line  is 
another  point  of  its  perspective  ;  whence,  the  perspective  of  a 
right  line  joins  its  vanishing  point  with  its  trace. 

It  is  customary  to  use  the  vertical  plane  as  the  perspective 
plane,  the  object  being  placed  in  the  second  angle  ;  this  brings 
the  two  projections  of  the  object  and  the  perspective  to- 
gether, as  in  Fig.  6,  PL  XXV.,  which  is  objectionable.  Dif- 
ferent methods  may  be  used  to  prevent  this,  but  the  most 
convenient  is  by  supposing  the  horizontal  plane  revolved 
180°,  so  as  to  bring  the  plan  of  the  object  in  front  of  V, 
and  then  instead  of  using  visual  rays,  to  make  use  of  aux- 
iliary lines  called  perpendiculars  and  diagonals,  by  which 
method  the  vertical  projection  of  the  object  is  not  necessary. 

It  is  evident  that  if  two  lines  intersect  at  a  point  in  space, 
their  perspectives  will  intersect  in  the  perspective  of  the 
point /  so  that  if  we  pass  any  two  lines  through  a  point  and 
find  their  perspectives,  their  intersection  will  be  the  perspec- 
tive of  the  point. 

The  two  lines  most  convenient  to  use  are  a  perpendicular 
and  a  diagonal. 

A  perpendicular  is  a  line  perpendicular  to  the  perspective 
plane,  and  vanishes,  as  we  have  seen,  in  the  centre  of  the  picture. 

A  diagonal  is  a  horizontal  line,  making  an  angle  of  45° 
with  the  perspective  plane. 

A  diagonal  being  a  horizontal  line  its  vanishing  point  is 
on  the  horizon,  and  since  it  makes  an  angle  of  45°  with  Y, 
the  distance  of  the  vanishing  point  from,  the  centre  of  the 
picture  is  equal  to  th-e  distance  of  the  point  of  sight  in  front 
of  V.  This  is  shown  in  Fig  2,  PL  XXVI.,  cc'  being  the  point 
of  sight,  and  c'd  the  horizon ;  to  find  the  vanishing  point  of 
diagonals  draw  through  cc'  the  line  ch — c'd,  parallel  to  H, 
and  making  an  angle  of  4:5°  with  Y ;  it  pierces  Y  at  <£,  the 
vanishing  point  of  all  diagonals  parallel  to  ch — c'd.  It  is 
evident  that  c/d=wh=cw.  As  diagonals  may  be  drawn 
either  to  the  right  or  left,  there  are  two  vanishing  points  of 
diagonals,  as  at  d  and  dl  (Fig.  2). 

Prob.  4.     (Fig.  8,  PL  XXY.)     To  construct  the  perspectiw 


162  INDUSTRIAL   DRAWING. 

of  a  regular  hexagon,  by  means  of  diagonals  and  perpen- 
diculars. The  hexagon  is  situated  in  H,  at  the  distance  nn' 
behind  Y.  The  horizontal  plane  has  been  revolved  180°,  so 
that  the  plan  of  the  hexagon  comes  in  front  of  Y,  while  the 
horizontal  projection  of  the  point  of  sight  at  the  same  time 
revolves  to  c,  behind  Y ;  c'  is  the  centre  of  the  picture,  <2  and 
di  the  vanishing  points  of  diagonals,  found  by  making  c'd  and 
c'di  equal  to  cw. 

To  find  the  perspective  of  the  point  e,  draw  through  it  the 
perpendicular  eef,  e'c'  is  its  perspective  ;  also  draw  through  e 
the  diagonal  ez,  zd^  is  its  perspective ;  the  point  o  in  which 
these  intersect  is  the  perspective  of  e  ;  g'  and  r  are  the  traces 
of  the  perpendicular  and  diagonal  through  g  •  g'c'  and  rd\  are 
their  perspectives,  and  I  is  the  perspective  of  g  /  p  is  the  per- 
spective of  f\  s  of  b  /  h  of  a  /  and  m  of  n. 

Remark.  In  Fig.  8  diagonals  are  drawn  in  both  directions, 
and  it  is  seen  that  either  diagonal  with  the  perpendicular 
gives  the  perspective,  or  that  two  diagonals  without  the  per- 
pendicular are  sufficient. 

Those  diagonals  which  in  plan  are  drawn  to  the  right,  van- 
ish to  the  left  of  the  centre  of  the  picture,  while  those  drawn 
to  the  left  in  plan,  vanish  to  the  right  of  the  centre  ;  this  comes 
from  having  revolved  the  plan  180°. 

Prob.  5.  (Fig.  9,  PI.  XXY.)  To  construct  the  perspective 
of  a  pavement  made  up  of  squares.  Let  abef  represent  the 
plan  of  the  pavement,  the  squares  being  set  with  their  sides 
diagonally  to  Y  ;  c'  is  the  centre  of  the  picture,  d  and  d^  the 
vanishing  points  of  diagonals. 

As  the  sides  of  the  squares  are  diagonal  lines,  their  perspec- 
tives will  join  their  traces  and  the  vanishing  points  of  diago- 
nals ;  produce  mn  to  0,  od\  is  the  perspective  of  mo  ;  ad  is 
the  perspective  of  ae  /  the  perspectives  of  the  other  edges  are 
similarly  found,  and  their  intersections  will  give  the  perspec- 
tives of  the  squares  ;  af  and  be,  being  perpendiculars,  vanish 
ate'. 

Prob.  6.  (Fig.  1,  PI.  XXYI.)  To  find  the  perspective  of  a 
cube.  Let  abih,  a'h'f'e'  be  the  projections  of  the  cube,  which 
is  placed  with  its  face  parallel  to  Y,  and  at  the  distance  ae' 
behind  Y. 


LINEAR  PERSPECTIVE.  163 

The  lines  e'c'  and/V  are  the  indefinite  perspectives  of  the 
lower  edges  of  the  cube,  which  are  horizontally  projected  in 
ab  and  hi  /  draw  the  diagonals  gd^  nd\,  md^  and  we  shall  ob- 
tain the  points  r,  o,  p,  z  ;  at  these  points  erect  perpendiculars 
and  limit  them  by  the  lines  a'c'  and  h'c'. 

Remark.  In  this  figure  the  vertical  projection  of  the  cube 
is  given,  but  it  evidently  is  not  necessary  in  order  to  construct 
the  perspective  ;  it  is  sufficient  to  know  the  height  of  the  cube, 
since  perpendiculars  and  diagonals  passed  through  points  in 
the  upper  base  of  the  cube  would  pierce  V  somewhere  in  the 
vertical  trace  of  the  plane  of  the  top,  as  they  are  horizontal 
lines,  and  are  in  that  plane. 

Prob.  7.  (Fig.  8,  PI.  XXYI.)  To  find  the  perspective  of 
a  vertical  hexagonal  prism.  The  prism  is  placed  with  one 
face  in  Y ;  the  line  rig'  is  the  vertical  trace  of  the  plane  of 
the  top  of  the  prism.  First,  construct  the  perspective  of  the 
lower  base  according  to  Prob.  4  ;  then  construct  the  perspec- 
tive of  the  upper  base,  remembering  that  the  traces  of  the 
perpendiculars  and  diagonals  passed  through  points  in  the 
upper  base,  will  be  in  the  line  n'g'.  Connect  the  two  bases 
by  vertical  lines  to  complete  the  prism. 

Prob.  8.  (Fig.  6,  PL  XXVI.)  To  find  the  perspective  of 
a  square  pillar  resting  upon  a  pedestal.  Let  dbhe,  a'e'ea  be 
the  projections  of  the  pedestal,  placed  with  its  face  inV; 
Tdpl  is  the  horizontal  projection  of  the  pillar;  k'l'  is  the  ver- 
tical trace  of  the  plane  of  the  top  of  the  pillar. 

First,  construct  the  perspective  of  the  pedestal  in  the  same 
way  that  the  perspective  of  the  cube  was  found  in  Prob.  6. 
The  face  aa'e'e  is  its  own  perspective,  as  it  is  in  V. 

To  find  the  perspective  of  the  pillar,  construct  first  the  per- 
spective of  the  lower  base  ;  draw  a  perpendicular  and  diago- 
nal through  the  point  Iri,  ri  and  e'  are  the  traces  of  these 
lines,  and  ric'  and  e'd\  their  perspectives  ;  their  intersection  is 
the  perspective  of  the  point  In' ;  g'  is  the  trace  of  the  diago- 
nal through p,  and  g'd\  its  perspective;  the  point  o  in  which 
this  intersects  ric'  is  the  perspective  of  the  point pn' ;  in  the 
same  way  the  remaining  points  of  the  base  can  be  found. 

The  perspective  of  the  upper  base  of  the  pillar  might  be 
found  in  the  same  way,  or  by  erecting  perpendiculars  at  the 


164:  INDUSTRIAL   DEALING. 

four  points  already  found,  and  limiting  them  by  the  lines  k'c' 
and  I'd. 

Prob.  9.  (Fig.  3,  PI.  XX  VI.)  To  find  the  perspective  of 
a  square  pyramid.  Let  abde  be  the  plan  of  the  pyramid ; 
hv'  is  the  height  of  the  vertex  above  the  base.  Find  first  the 
perspective  of  the  base ;  e  is  its  own  perspective,  as  it  is  in 
V ;  m  is  the  perspective  of  a  ;  em  the  perspective  of  ea  /  ep 
the  perspective  of  ed,  etc. 

To  obtain  the  perspective  of  the  vertex,  find  o,  the  perspec- 
tive of  its  horizontal  projection  ;  at  o  erect  a  perpendicular 
until  it  meets  v'c',  this  gives  n,  the  perspective  of  the  vertex ; 
joining  n  with  the  corners  of  the  base  completes  the  per- 
spective. 

Prob.  10.  (Fig.  5,  PI.  XXVI.)  To  find  the  perspective 
of  a  square  pyramid  resting  upon  a  pedestal.  Let  abeh  be 
the  plan  of  the  pedestal ;  r'w'  is  the  vertical  trace  of  the  upper 
base ;  the  edge  aa'  is  in  V  and  is  part  of  the  perspective ;  z 
is  the  perspective  of  the  point  W ;  <K  of  hh't  etc. 

After  completing  the  perspective  of  the  pedestal,  find  the 
perspective  of  the  base  of  the  pyramid  ;  n'c'  and  p'd  are  the 
perspectives  of  a  perpendicular  and  diagonal  through  n,  and 
o  is  the  perspective  of  n  ;  in  the  same  way  the  other  corners 
of  the  base  can  be  found. 

To  find  the  perspective  of  the  vertex,  draw  through  v  a 
perpendicular  and  diagonal ;  their  traces  are  v'  and  p'  and 
their  perspectives  v'o'  and  p'd  j  s  is  the  perspective  of  the 
vertex. 

Prob.  11.  (Fig.  7,  PI.  XXVI.)  To  find  the  perspective 
of  an  hexagonal  prism,  whose  axis  is  parallel  to  H  and  in- 
clined to  V. 

The  prism  rests  with  one  face  in  H  ;  aeJd  is  the  plan ;  m'p' 
is  the  trace  upon  V  of  the  plane  of  the  upper  face  of  the 
prism ;  t'n'  is  the  trace  upon  V  of  the  horizontal  plane  which 
contains  the  edges  projected  in  al  and  ek  ;  ccf  is  the  point  of 
sight. 

As  the  edges  al,  bh,  etc.,  of  the  prism  are  parallel  lines,  and 
are  so  situated  that  they  pierce  V  within  the  drawing,  the 
most  convenient  way  to  find  their  perspectives  is  by  joining 
their  traces  with  their  vanishing  point.  To  find  the  vanish- 


LDTEAE   PERSPECTIVE.  165 

ing  point,  draw  cw—c'w'  parallel  to  at,  bh,  etc. ;  this  pierces 
V  at  w',  the  vanishing  point  of  the  edges  of  the  prism. 

The  edge  al  pierces  Y  at  «',  and  a'w'  is  its  indefinite  per- 
spective ;  this  is  limited  at  s  by  the  diagonal  t'd;  q'  is  the 
trace  of  bh,  when  an  edge  of  the  upper  face,  and  q'w'  is 
its  indefinite  perspective;  this  is  limited  by  the  diagonals 
m'd  and  o'd  ;  qw'  is  the  indefinite  perspective  of  bh,  when  an 
edge  of  the  lower  face,  and  is  limited  by  the  diagonals  md  and 
od ;  in  the  same  way  the  perspective  of  the  other  edges  can 
be  found.  The  method  of  completing  the  perspective  of  the 
prism  will  be  apparent. 

Construct  the  perspective  when  the  axis  of  the  prism  ie 
parallel  to  both  planes  of  projection. 

Prob.  12.  (Fig.  1,  PI.  XXVII.)  To  find  the  perspective 
of  a  circle.  Let  aebk  be  the  circle  situated  in  the  horizontal 
plane.  Find  the  perspectives  of  the  squares  aebk  and  hmno  ; 
this  will  give  eight  points  of  the  perspective  of  the  circle ;  the 
curve  should  be  tangent  to  the  sides  of  the  circumscribed 
square  at  the  points  xzyv. 

Prob.  13.  (Fig.  2,  PI.  XXVII.)  To  find  the  perspective 
of  a,  circle  when  it  is  perpendicular  to  both  H  and  V.  Let 
ek  and  a'b'  be  the  projections  of  the  circle ;  ad"a"e"  repre- 
sents the  circle  with  inscribed  and  circumscribed  squares, 
when  revolved  about  the  point  ab'  into  H. 

To  find  the  perspective  of  the  circumscribed  square,  draw 
a'c'  and  b'c'  j  these  are  the  indefinite  perspectives  of  the 
upper  and  lower  edges  ;  draw  the  diagonals  kp  and  eg,  and 
find  their  perspectives  ;  at  r  and  I  erect  perpendiculars,  and 
they  will  be  the  perspectives  of  the  vertical  edges  of  the 
square. 

The  perspective  of  the  inscribed  square  can  be  found  in  the 
same  way ;  b'o'  is  equal  to  oo"  and  b'h'  to  oh. 

Prob.  14.  (Fig.  5,  PI.  XXVII.)  To  find  the  perspective 
of  a  vertical  cylinder.  The  cylinder  is  tangent  to  V;  xy  is 
the  vertical  trace  of  the  plane  of  the  top.  Construct  the  per- 
spectives of  the  two  bases  by  Prob.  12.  Vertical  lines  tan- 
gent to  the  two  curves  will  be  the  extreme  elements  of  the 
cylinder. 

Remark.     In  this  problem,  and  some  of  the  others,  the  per- 


166  INDUSTRIAL  DRAWING. 

epecti-ves  appear  somewhat  unnatural ;  this  is  owing  to  the 
point  of  sight  being  taken  too  near  the  object ;  with  more 
room  this  can  be  remedied. 

Prob.  15.  (Fig.  4,  PL  XXYII.)  To  find  the  perspective 
of  a  cylinder  whose  axis  is  perpendicular  to  Y.  Let  abef 
be  the  cylinder  resting  upon  H.  As  the  ends  of  the  cylinder 
are  parallel  to  Y,  their  perspectives  will  be  circles,  and  it  will 
only  be  necessary  to  locate  their  centres  and  determine  the 
length  of  their  radii ;  o'c'  is  the  indefinite  perspective  of  the 
axis  of  the  cylinder  ;  o'x  being  equal  to  ob  ;  n'd\  is  the  per- 
spective of  the  diagonal  through  o  ;  hence  r  is  the  perspec- 
tive of  the  centre  of  the  front  end,  and  rv,  the  perspective  of 
0'a?,  is  the  radius ;  w  is  the  centre  of  the  back  end,  and  ws 
the  radius ;  lines  drawn  from  c'  tangent  to  these  two  circles 
will  be  the  extreme  elements  of  the  cylinder. 

The  cylinder  is  represented  with  a  square  hole  running 
through  it ;  the  perspective  of  the  hole  is  left  for  the  student 
to  construct. 

Construct  the  perspective  of  a  cylinder  with  a  circular  hole 
running  through  it. 

Prob.  16.  (Fig.  3,  PI.  XXYII.)  To  find  the  perspective 
of  a  rectangular  block  with  a  semicircular  top.  The  lower 
part  of  the  block  is  square,  with  a  square  opening  in  it ;  the 
top  is  semicircular  with  a  triangular  opening  ;  bke  represents 
the  projection  of  the  top  when  revolved  about  the  horizontal 
diameter  be  of  the  semicircle  until  parallel  to  H  ;  cc'  is  the 
centre  of  the  picture  ;  d  and  d±  the  vanishing  points  of  diago- 
nals, and  also  the  vanishing  points  of  the  edges  of  the  block, 
as  it  is  placed  diagonally  to  Y. 

The  edge  ap  is  equal  to  aft  and  since  it  is  in  Y,  it  is  its  own 
perspective  ;  pd\  and  ad\  are  the  indefinite  perspectives  of  the 
upper  and  lower  edges  of  the  block ;  using  the  diagonal  ud 
we  obtain  oz  as  the  perspective  of  the  vertical  edge,  which  is 
horizontally  projected  inf. 

To  obtain  the  perspective  of  the  opening,  make  al  and  pq 
each  equal  to  an  /  draw  the  lines  qd\  and  ld\  /  these  are  the 
indefinite  perspectives  of  the  top  and  bottom  edges  of  the 
hole;  the  sides  are  obtained  by  erecting  perpendiculars 
through  the  points  t  and  r. 


LINEAB  PERSPECTIVE.  167 

The  completion  of  the  rest  of  the  perspective  of  the  lower 
part  of  the  block  is  evident  from  the  figure. 

To  construct  the  perspective  of  the  upper  part,  makep&'= 
yk;  ph'  =  mh  ;  pv'  =  yv  ;  draw  the  lines  k'd^  h'd^  v'd^ 
where  these  meet  perpendiculars,  erected  through  t,  g,  r,  will 
be  points  of  the  perspective ;  in  the  same  way  other  points 
may  be  found  if  necessary.  The  completion  of  the  perspec- 
tive is  left  for  the  student. 

Construct  the  perspective  when  the  block  is  placed  so  toat 
its  edges  will  not  vanish  at  d  and  d^ 


168  INDUSTRIAL   DRAWING. 


CHAPTER  XIII. 

ON  THE  MANNER  OF  REPRESENTING  SOME  OF  THE  MORE  USUAL 
PRINCIPAL  ARCHITECTURAL  ELEMENTS. 

Roof  Truss.  This  term  is  applied  to  the  main  framea 
placed  vertically  on  the  top  of  the  walls  of  a  building  to  sup- 
port the  roof  covering. 

The  most  simple  form  of  truss  (PI.  VIII.  Fig.  99)  consists 
of  a  horizontal  beam,  A,  resting  on  the  walls,  termed  the  tie- 
beam  ;  of  two  beams,  B,  suitably  inclined  for  the  intended 
slope  of  the  roof,  termed  the  main  rafters  /  of  a  vertical  post, 
C,  attached  to  the  tie-beam  at  its  centre,  and  against  which, 
at  its  top,  the  rafters  rest,  termed  the  king-post,  and  two 
inclined  pieces,  M,  resting  against  the  bottom  of  the  king-post, 
and  at  their  upper  ends  against  the  middle  of  the  rafters, 
termed  struts,  or  braces. 

In  small  buildings  the  king-post  and  struts  are  sometimes 
left  out;  and  in  some  cases  the  tie-beam  also. 

In  large  trusses  (PI.  VIII.  Fig.  100)  two  vertical  posts  D, 
termed  queen-posts,  replace  the  single  king-post;  auxiliary 
rafters,  E,  are  placed  under  the  main  rafters,  and  rest  against 
a  horizontal  beam,  F,  termed  a  straining  beam,  placed  above 
the  tie-beam. 

The  queen-posts  are  usually  of  two  pieces  each,  and  receive 
the  other  beams  between  them  into  notches  cut  in  each  piece ; 
and,  in  some  cases,  a  king-post  connects  the  middle  of  the 
straining  beam  with  the  top  of  the  rafters. 

On  the  main  rafters  are  placed  beams,  G,  laid  horizontal^, 
termed  purlins  /  these  support  other  inclined  beams,  I,  termed 
long  rafters,  to  which  the  boards,  slate,  metal,  etc.,  of  the 
roof  covering  are  attached.  The  top  purlin  N,  resting  on  the 
top  of  the  main  rafters,  is  termed  the  ridge  beam;  the  bottom 


AECHTTECTUEAL  ELEMENTS.  169 

one,  0,  on  which  tlje  ends  of  the  long  rafters  rest,  fhe  pole- 


Drawing  of  a  rooft-uss.  We  commence  the  drawing  of 
the  truss  by  constructing  the  triangle  abc  formed  by  the 
top  line  of  the  tie-beam  and  the  inner  lines  of  the  main 
rafters.  Next  draw  the  centre  lines  d — e  and  g — A  of  the 
king  and  queen-posts ;  next  the  top  line,  m — nt  of  the  strain- 
ing beam. 

Having  thus  made  an  outline  sketch  of  the  general  form, 
proceed  by  putting  in  the  other  lines  of  the  different  beams 
in  their  order. 

To  show  the  connection  of  the  queen-posts  with  the  tie- 
beam,  etc.,  a  longitudinal  section  (PI.  VIII.  Fig.  101)  is 
given,  which  may  be  drawn  on  a  larger  scale,  if  requisite. 
Also  a  drawing  to  a  larger  scale  is  sometimes  made,  to  show 
the  connection  between  the  bottom  of  the  rafters  and  the 
tie-beam. 

Remarks.  As  it  is  usual  in  drawings  of  simple  frames 
like  the  above  to  give  but  one  projection,  showing  only  the 
cross  dimensions  of  the  beams  in  one  direction,  the  dimen- 
sions in  the  other  are  witten  either  above,  or  alongside  of  the 
former.  When  above,  a  short  line  is  drawn  between  them  ; 
when  alongside,  the  sign  x  of  multiplication  is  placed  between 
them.  The  better  plan  is  to  write  the  number  of  the  cross 
dimension  that  is  projected  in  the  usual  way,  and  to  place  the 
other  above  it  with  a  short  line  between,  as  in  PI.  XI.  Fig.  c, 
that  is,  8  inches  by  9  inches. 

Columns  and  Entablatures.  In  making  drawings  of  these 
elements,  it  is  usual  to  take  the  diameter  of  the  column  at  the 
base,  and  divide  it  into  sixty  equal  pails,  termed  minutes  / 
the  radius  of  the  base  containing  thirty  of  those  parts,  and 
termed  a  module,  being  taken  as  the  unit  of  measure,  the 
fractional  parts  of  which  are  minutes.  For  building  purposes 
the  actual  dimensions  of  the  various  parts  would  be  expressed 
in  feet  and  fractional  parts  of  a  foot. 

To  commence  the  drawing,  three  parallel  lines  (PI.  IX.  Fig. 
102)  are  drawn  on  the  left-hand  side  of  the  sheet  of  paper. 
The  middle  space,  headed  S,  is  designed  to  express  the 
heights,  or  distances  apart  vertically  of  the  main  divisions; 


170  INDUSTRIAL   DBAWINQ. 

that  on  the  left,  headed  E,  is  for  the  heights  of  the  subdi 
visions  of  the  mam  portions ;  and  that  on  the  right,  headed 
T,  is  for  what  are  termed  the  projections,  that  is,  the  distances 
measured  horizontally  between  the  centre  line,  or  axis  of  the 
column,  and  the  parts  which  project  beyond  the  axis. 

At  a  suitable  distance  on  the  right  of  these  lines  another 
parallel,  X — Y,  is  drawn  for  the  axis  of  the  column.  At  some 
suitable  point,  towards  the  bottom  of  the  sheet,  a  perpendicu- 
lar is  drawn  to  the  axis,  and  prolonged  to  cut  the  parallels 
to  it.  This  last  line  is  taken  as  the  bottom  line  of  the  base 
of  the  column.  From  this  line  set  off  upwards — 1st,  the 
height  of  the  base ;  2d,  that  of  the  shaft  of  the  column ;  3d, 
that  of  the  capital  of  the  column ;  4th,  the  three  divisions  of 
the  entablature ;  and  through  these  points  draw  parallels  to 
the  bottom  line. 

Commencing  now  at  the  top  horizontal  line  set  off  along  it 
from  the  axis,  to  the  right,  the  distance  a — b,  equal  to  the 
projection  of  the  point  b  /  in  the  same  way  the  projections  of 
the  successive  points,  in  their  order  below,  as  d,f,  etc.  Hav- 
ing marked  these  points  distinctly,  to  guide  the  eye  in  draw- 
ing the  other  lines  for  projections,  commence  by  setting  off 
accurately,  from  the  top  downwards,  the  heights  of  the  respec- 
tive subdivisions  along  the  space  headed  It.  These  being  set 
off  draw  parallels,  through  the  points  set  off,  to  the  horizon- 
tals, commencing  at  the  top,  and  guiding  the  eye  and  hand 
by  the  points  5,  d,  etc.,  in  order  not  to  extend  the  lines  un- 
necessarily beyond  the  axis. 

Having  drawn  the  horizontals,  proceed  to  set  off  upon  them 
their  corresponding  projections ;  which  done,  connect  the 
horizontal  lines  by  right  lines,  or  arcs  of  circles,  as  shown  in 
the  figure. 

Mouldings.  The  mouldings  in  architecture  are  the  portions 
formed  of  curved  surfaces.  The  outlines,  or  profiles  of  those 
in  most  common  use  in  the  Koman  style,  are  shown  in  (PI. 
IX.  Fig.  103),  they  consist  of  either  a  single  arc  of  a  circle, 
which  form  what  are  termed  simple  mouldings ;  or  of  two  or 
more  arcs,  termed  compound  mouldings.  The  arcs  in  the 
Plate  are  either  semicircles,  as  in  the  torus,  Etc.,  or  quadrants 
of  a  circle,  as  in  the  cavetto,  scotia,  etc.  The  manner  of 


ARCHITECTURAL   ELEMENTS.  171 

constructing  these  curves  is  explained  (PL  III.  Figs.  39,  40, 
etc.). 

The  entire  outline  to  the  right  of  the  axis  is  termed  a 
profile  of  the  column  and  entablature. 

RemarT&s.  In  setting  off  projections,  those  of  the  parts 
above  the  shaft  are  sometimes  estimated  from  the  outer  point 
of  the  radius  of  the  top  circle  of  the  shaft ;  and  those  below 
it  from  the  outer  point  of  the  lower  radius ;  but  the  method 
above  explained  is  considered  the  best,  as  more  uniform. 

Where  the  scale  of  the  drawing  is  too  large  to  admit  of  the 
entire  column  being  represented,  it  is  usual  to  make  the 
drawing,  as  shown  in  the  figure,  a  part  of  the  shaft  being 
supposed  to  be  removed. 

The  outline  of  the  sides  of  the  shaft  are  usually  curve  lines, 
and  constructed  as  follows : — Having  drawn  a  line  o— -p  (PI. 
IX.  Fig.  104)  equal  to  the  axis,  and  the  lines  of  the  top  and 
bottom  diameters  being  prolonged,  set  off  on  the  latter  their 
respective  radii  o — b,  andj? — a.  From  a  set  off  a  distance 
to  the  axis  a — c,  equal  to  o — b,  the  lower  radius ;  and  prolong 
a — c,  to  meet  the  lower  diameter  prolonged  at  Z.  From  the 
point  Z  draw  lines  cutting  the  axis  at  several  points,  as  d,  e,f, 
etc.  From  these  points  set  off  along  the  lines  Z — d,  etc.,  the 
lengths  d — m,  e — n,  &c.,  respectively  equal  to  a — c,  or  the 
lower  radius ;  the  points  m,  n,  etc.,  joined,  will  be  the  outline 
of  the  side  of  the  column. 

Arches.  (PI.  YIII.  Fig.  105.)  The  arch  of  simplest  form, 
and  most  usual  application  in  structures,  is  the  cylindrical, 
that  is,  one  of  which  the  cross  section  is  the  same  throughout, 
and  upon  the  interior  surface  of  which,  termed  the  soffit  of 
the  arch,  right  lines  can  be  drawn  between  the  two  ends  of 
it.  The  cross  sections  of  most  usual  form  are  the  semicircle, 
an  arc  of  a  circle,  oval  curves,  and  curves  of  four  centres. 

Eight  arch.  The  example  selected  for  this  drawing  is  the 
one  with  a  semicircular  cross  section,  the  elements  of  the 
cylinder  being  perpendicular  to  the  ends,  and  which  is  termed 
the  full  centre  right  arch. 

We  commence  this  drawing  by  constructing,  in  the  first 
place,  the  elevation  of  the  face,  or  the  front  view  of  the  end 
of  the  arch.  Having  drawn  a  ground  line  G — Z,  set  off 


172  INDUSTRIAL   DRAWING. 

from  any  point,  as  a,  a  distance  a — 5,  equal  to  the  diametei 
of  the  semicircle  of  the  cross  section ;  erecting  perpendicu- 
lars at  a  and  5,  set  off  on  them  the  equal  distances  a — A,  and 
5 — B,  for  the  height  of  the  lower  lines  of  the  arch  above  the 
horizontal  plane  of  projection ;  join  A  and  S,  and  on  it  de- 
scribe the  semicircle  ACS /  and  from  the  same  centre  O 
another  parallel  to  it,  with  any  assumed  radius  OD.  Divide 
the  semicircle  A  OB  into  any  odd  number  of  equal  parts,  as 
five,  for  example  ;  and  from  the  centre  draw  the  radii  Om, 
etc.,  through  the  points  of  division  mt  etc.,  and  prolong  them 
to  m',  etc.,  on  the  outer  semicircle.  From  the  points  J),  m', 
n',  etc.,  draw  the  vertical  lines  D — d,  m' — c,  n' — #,  etc.,  to 
intersect  the  horizontal  lines  drawn  through  tp,'  n',  etc.  The 
pentagonal  figures  Bmm'dD,  etc.,  thus  formed  will  be  the 
form  of  the  arch  stones  of  the  face  of  the  arch.  The  top 
stone,  at  O,  is  termed  the  fay-stone,  its  vertical  side  n' — a  is 
taken  at  pleasure. 

Having  obtained  the  shapes  of  the  face  stones,  we  next 
proceed  to  draw  those  of  the  face  of  the  wall  contiguous  to 
them.  The  manner  of  combining  the  stones  of  these  two 
parts  is  purely  arbitrary.  The  only  rule  is  so  to  combine 
the  horizontal  lines  of  the  two  as  to  present  a  pleasing  archi- 
tectural effect  to  the  eye.  The  ifsual  method,  for  obtaining 
this  result,  is  to  bring  the  horizontal  lines  of  the  face  of  the 
wall,  on  each  side  of  the  arch,  to  be  on  the  prolongations  of 
those  A — Z>,  m' — d,  etc.,  of  the  arch ;  and  so  to  divide  the 
vertical  distances  D — d,  m' — c,  etc.,  that  the  distances  between 
the  intermediate  horizontals  shall  be  as  nearly  equal  as  prac- 
ticable. The  distances  between  the  horizontals  of  the  face 
of  the  wall,  below  the  arch,  are  usually  the  same,  and  equal 
to  those  just  above. 

If  the  walls  on  which  the  arch  rests  are  built  for  the  sole 
purpose  of  supporting  it  they  are  termed  abutments.  The 
portions  of  these,  here  shown  in  front  view,  are  termed  the 
ends,  or  heads  of  the  abutments ;  the  portions  projected  in 
A — a,  and  B — 5,  and  lying  below  the  suffit  of  the  arch,  are 
termed  the  sides  or  faces. 

The  lines  projected  in  A,  and  B^  along  which  the  soffit 
joins  the  faces  of  the  abutments,  are  termed  the  springing 


AEOHTTEOTTJEAL  ELEMENTS.  173 

lines  of  the  arch.  The  right  line  projected  in  0,  and  parallel 
to  the  springing  lines,  is  termed  the  axis  of  the  arch.  The 
right  lines  of  the  soffit  projected  in  m,  n,  etc.,  are  termed  the 
soffit  edges  of  the  coursing  joints  of  the  arch ;  the  lines  m — w', 
n — n't  etc.,  the  face  edges  of  the  same. 

To  construct  a  longitudinal  section  of  the  arch  by  a  vertical 
plane  through  the  axis  of  which  the  trace  on  the  face  is  the 
line  M—N,  commence  by  drawing  a  line  V — B"  parallel 
to  b — B,  and  at  any  convenient  distance  from  the  front  eleva- 
tion ;  from  V  set  off  along  the  ground  line  the  distance  V — &', 
equal  to  that  between  the  front  and  back  faces,  or  the  length 
of  the  arch ;  from  V  draw  V—B'  parallel  to  V'—B",  and 
prolong  upwards  these  two  lines.  The  rectangle  b'B'B"b" 
will  be  the  projection  of  the  face  of  the  abutment  on  the  plane 
of  section  ;  the  line  b' — B ',  corresponding  to  that  b — B,  etc. 
Drawing  the  horizontal  lines  B" — £',  m" — m'",  etc.,  at  the 
same  height  above  the  ground  line  as  the  respective  points  B, 
m,  etc.,  they  will  be  the  projections  of  the  soffit  edges  of  the 
coursing  joints.  The  half  of  the  soffit  on  the  right  of  the 
plane  of  section  M — .ZV7",  is  projected  into  the  rectangle 
B '  G'C"B".  The  arch  stones  &,  &,  etc.,  forming  the  key  of 
the  arch,  are  represented  in  section,  the  two  forming  the  ends 
of  greater  depth  than  those  intermediate,  as  is  very  often  done. 
That  is,  the  key- stones  at  the  ends,  and  the  end  walls  of  the 
abutments,  are  built  up  higher  than  the  interior  masonry  be- 
tween them ;  the  top  of  this  last  being  represented  by  the 
dotted  line  o—p  in  the  elevation  and  the  full  line  o—p'  on 
the  cross  section. 

The  arch  stones  running  through  from  one  end  to  the 
other,  and  projected  between  any  two  soffit  edges,  as  B" — B', 
and  m" — m"'t  are  termed  a  string  course.  The  contiguous 
stones  running  from  one  springing  line  to  the  other,  as  those 
projected  in  &,  &',  &",  &c.,  are  termed  ring  courses.  The 
lines,  of  which  those  r — s  are  the  projections,  are  the  soffit 
edges  of  the  joints,  termed  heading  joints,  between  the  stones 
of  the  string  courses.  These  edges  in  one  course  alternate 
with  those  of  the  courses  on  either  side  of  it. 

The  cross  section  on  R — S  requires  no  particular  explana- 
tion. From  its  conventional  lines,  it  will  be  seen  that  the 


174  INDUSTRIAL  DRAWING. 

intention  is  to  represent  the  soffit  and  faces  of  the  arch,  and 
its  abutments,  as  built  of  cut  stone,  and  the  backing  of  rubble 
stone. 

Remarks.  In  the  drawings  just  made,  it  will  be  observed 
that  no  plan,  or  horizontal  projections,  were  found  to  be 
necessary  ;  and  that  a  perfectly  clear  idea  is  given  of  the 
forms  and  dimensions  of  all  the  parts  by  means  of  the  eleva- 
tions and  sections  alone.  A  previous  study  of  the  particular 
object  to  be  represented  will  very  frequently  lead  to  like  re- 
sults, wherein  a  few  views,  judiciously  chosen,  will  serve  all 
the  objects  of  a  drawing. 

The  methods  of  arranging  the  vertical  and  horizontal  lines 
of  the  arch  stones  of  the  head,  with  those  of  the  abutments, 
often  present  problems  of  some  intricacy,  demanding  both 
skill  and  taste  on  the  part  of  the  draftsman,  so  to  combine 
them  as  to  produce  a  pleasing  architectural  appearance,  and 
yet  not  interfere  with  other  essential  conditions.  An  exam- 
ple of  this,  being  the  case  of  a  segment  arch,  is  given  in  PI. 
VIII.  Fig.  106. 


PROBLEMS   OF   MECHANISM.  175 


CHAPTER  XIV. 


Prdb.  114.  (PL  XIY.  Fig.  137.)  To  construct  the  projec 
tions  of  a  cylindrical  spur  wheel. 

The  wheel  work  employed,  in  mechanism,  to  transmit  the 
motion  of  rotation  of  one  shaft  to  another  (the  axis  of  the 
second  being  parallel  to  that  of  the  first),  usually  consists  of 
a  cylindrical  disk,  or  ring,  from  the  exterior  surface  of  which 
projects  a  spur  shaped  combination,  termed  teeth,  or  cogs,  so 
arranged  that,  the  teeth  on  one  wheel  interlocking  with  those 
of  the  other,  any  motion  of  rotation,  received  by  the  one 
wheel,  is  communicated  to  the  other  by  the  mutual  pressure 
of  the  sides  of  the  teeth.  There  are  various  methods  by 
which  this  is  effected ;  but  it  will  be  only  necessary  in  this 
place  to  describe  the  one  of  most  usual  and  simple  construc- 
tion, for  the  object  we  have  in  view. 

The  thickness  of  each  tooth,  and  the  width  of  the  space 
between  each  pair  of  teeth,  are  set  off  upon  the  circumference 
of  a  circle,  which  is  termed  the  pitch  line  or  pitch  circle.  The 
thickness  of  the  tooth  and  the  width  of  the  space  taken 
together,  as  measured  along  the  pitch  line,  is  termed  the  pitch 
of  the  tooth.  The  pitch  being  divided  into  eleven  equal  parts, 
five  of  these  parts  are  taken  for  the  thickness  of  the  tooth, 
and  six  for  the  width  of  the  space.  Having  given  the  radius 
o — m  of  the  pitch  circle,  and  described  this  circle,  it  must  first 
be  divided  into  as  many  parts,  each  equal  to  b—  A,  the  pitch  of 
the  teeth,  as  the  number  of  teeth.  Having  made  this  division, 
the  outline  of  each  tooth  may  be  set  out  as  follows  : — From 
the  point  A,  with  the  distance  h — b  as  a  radius,  describe  an 
arc  b — c,  outwards  from  the  pitch  circle  ;  having  set  off  b — e, 
the  thickness  of  the  tooth ;  from  the  point  k,  with  the  same 
radius,  describe  the  arc  e— /  To  obtain  the  apex  c—f,  of  the 
tooth,  which  may  be  either  a  right  line,  or  an  arc  described 
from  o  as  a  centre ;  place  this  arc  three-tenths  of  the  pitch 
fr— h  from  the  pitch  line.  The  sides  of  the  tooth,  within  the 


176  INDUSTRIAL   DRAWING. 

pitch  circle,  are  in  the  directions  of  radii  drawn  from  the 
points  b  and  e.  The  bottom  d — g  of  each  space  is  also  an  arc, 
described  from  o,  and  at  a  distance,  from  and  within  the 
pitch  circle,  of  four-tenths  of  the  pitch.  From  the  preceding 
construction,  the  outline  of  each  tooth  will  be  the  same  as 
abcfed;  and  that  of  each  space  fdgi.  The  curved  portions 
b— -c.  and  e—fof  each  tooth  are  termed  the  faces,  the  straight 
portions  a — &,  and  d — e,  the  flanks.  The  outline  here 
described  represents  the  profile  of  the  parts,  made  by  a  plane 
perpendicular  to  the  axis  of  the  cylindrical  surface,  to  which 
the  teeth  are  attached ;  which  surface  forms  the  bottom  of 
the  spaces.  The  breadth  of  the  teeth  is  the  same  as  the 
length  of  the  cylindrical  surface  to  which  they  are  attached. 

The  solid  to  which  the  teeth  are  attached  or  of  which  they 
form  a  portion  is  a  rim  of  the  same  material  as  the  teeth  ; 
and  this  rim  is  attached  to  a  central  boss,  either  by  arms,  like 
the  spokes  of  a  carriage  wheel,  or  else  by  a  thin  plate.  The 
former  is  the  method  used  for  large  wheels ;  the  latter  for 
small  ones. 

In  wooden  wheels,  the  cogs  are  let  into  the  rims,  by  holes 
cat  through  the  rims.  In  cast  iron  wheels,  the  rim  and  teeth 
are  cast  in  one  solid  piece.  When  the  latter  material  is 
used,  the  arms  and  central  boss  are  also  cast  in  one  piece 
with  the  rim  and  teeth  in  medium  wheels ;  but  in  large  sized 
ones,  the  wheel  is  cast  in  several  separate  portions,  which  are 
afterwards  fastened  together  and  to  the  arms,  &c. 

In  the  example  before  us,  we  shall  suppose,  for  simplifica- 
tion, that  the  rim  joins  directly  to  the  boss ;  the  latter  being 
a  hollotv  cylinder,  projecting  slightly  beyond  each  face  of  the 
wheel ;  the  diameter  of  the  hollow  being  the  same  as  that  of 
the  cylindrical  shaft  on  which  the  wheel  is  to  be  placed. 

Let  us  now  suppose  the  axis  of  the  wheel  horizontal  and 
perpendicular  to  the  vertical  plane.  In  this  position  of  the 
wheel,  the  outline  just  described  will  be  the  vertical  projec- 
tion of  the  wheel;  the  rectangle  ESTUike  horizontal  projec- 
tions of  the  teeth  and  rim ;  and  the  one  OO'P'P,  that  of  the 
boss,  which  projects  the  equal  distances  S—  0,  and  R — 0\ 
beyond  the  faces  R—  T,  and  S—Uof  the  wheel. 

The  faces  of  each  tooth,  the  surface  of  its  apex,  and  the 


PROBLEMS  OF  MECHANISM.  177 

bottom  of  each  space  are  all  portions  of  cylindrical  surfaces, 
the  elements  of  which  are  parallel  to  the  axis  of  the  wheel. 
The  horizontal  projections  of  the  edges  of  the  teeth  C"—  (7'and 
F"—F',  which  correspond  to  the  points  projected  in  c,  and^ 
as  well  as  those  of  the  spaces,  which  correspond  to  the  points 
a  and  d,  will  be  right  lines  parallel  to  0 — 0'.  Drawing  the 
projections  as  C—  C',  and  F—  F',  &c.,  of  these  edges,  we 
obtain  the  complete  horizontal  projection  of  the  wheel. 

Prob.  115.  (PI.  XIY.  Fig.  138.)  To  construct  the  projec- 
tions of  the  same  wheel  when  the  axis  is  still  horizontal  but  oblique 
to  the  vertical  plane. 

As  in  the  preceding  Probs.,  of  like  character  to  this,  the 
horizontal  projection  of  all  the  parts  will,  in  this  position  of 
the  axis,  be  the  same  as  in  the  preceding  case ;  and  the 
vertical  projections  will  be  found  as  in  like  cases.  The  pitch 
circle  and  other  circles  on  the  faces  of  the  wheel,  and  the 
ends  of  the  boss,  will  be  projected  in  ellipses ;  the  transverse 
axes  of  which  are  the  vertical  diameters  of  these  circles ;  and 
the  conjugate  axes  the  vertical  projections  of  the  correspond- 
ing horizontal  diameters.  The  vertical  projections  of  the 
edge?  of  the  teeth,  which  correspond  to  the  horizontal  pro- 
jections C—  C',  and  F — F',  &c.,  will  be  the  lines  c — c'",  and 
/—/'",  &c.,  parallel  to  the  projection  o'— o",  of  the  axis. 

Prob.  116.  (PI.  XY.  Fig.  139.)  To  construct  the  projections 
of  a  mitre,  or  beveled  wheel,  the  axis  of  the  wheel  being  horizontal, 
and  perpendicular  to  the  vertical  plane. 

The  spur  wheel,  we  have  seen,  is  one  in  which  the  teeth 
project  beyond  a  cylindrical  rim,  attached  to  a  central  boss 
either  by  arms,  or  by  a  thin  connecting  plate  ;  moreover  that 
portions  of  the  teeth  project  beyond  the  pitch  line,  or  circle, 
whilst  other  portions  lie  within  this  line.  Mitre,  or  beveled 
wheels,  are  those  in  which  the  teeth  are  attached  to  the 
surface  of  a  conical  rim ;  the  rim  being  connected  with  a 
central  boss,  either  by  arms,  or  a  connecting  plate.  In  the 
beveled  wheel  the  faces  of  the  teeth  project  beyond  an 
imaginary  conical  surface,  termed  the  pitch  cone,  whilst  the 
flanks  lie  within  the  pitch  cone.  The  faces  and  flanks  are 
conical  surfaces,  which  have  the  same  vertex  as  the  pitch 
cone  •  the  apex  of  each  tooth  is  either  a  plane,  or  a  conical 
12 


178  INDUSTRIAL   DRAWING. 

surface  which,  if  prolonged,  would  pass  through  the  venes 
of  the  pitch  cone ;  and  the  bottom  of  the  space  between  each 
tooth  is  also  a  portion  of  a  cone,  or  a  plane  passing  through 
the  same  point. 

The  ends  of  the  teeth  and  the  rim  lie  on  conical  surfaces, 
the  elements  of  which  are  perpendicular  to  those  of  the  pitch 
cone,  and  have  the  same  axis  as  it. 

Before  commencing  the  projections,  it  will  be  necessary  to 
explain  how  the  teeth  are  set  out,  as  well  as  the  rim  from 
which  they  project.  Let  F  (Figs.  A,  B)  be  the  vertex  of  the 
pitch  cone ;  V — o  its  axis ;  and  Vmn  its  generating  triangle. 
At  n  and  TO,  drawing  perpendiculars  to  V- — n  and  V- — TO,  let 
the  point  v,  where  they  meet  the  axis  prolonged,  be  the 
vertex  of  the  cone  that  terminates  the  larger  ends  of  the 
teeth  and  rim.  Setting  off  the  equal  distances  TO — TO'  and 
n — w',  and  drawing  TO' — v'  and  n' — v'  respectively  parallel  to 
TO — v  and  n — v,  let  v'  be  the  vertex  of  the  cone,  and  v'm'n' 
its  generating  triangle  that  forms  the  other  end  of  the  teeth 
and  rim. 

Having  by  Prob.  114  developed  the  cone,  of  which  v  is 
the  vertex  (Fig.  B]  and  TO — n  the  diameter  of  the  circle  of  its 
base ;  set  off  upon  the  circle,  described  with  the  radius 
v—  n,  the  width  n — b  of  the  space  between  the  teeth,  together 
with  the  thickness  b — e  of  the  teeth,  as  in  the  preceding 
Prob.  115,  and  construct  the  outline  of  each  tooth  and  space 
as  in  those  cases ;  next  set  off  a — t,  equal  to  a — b,  for  the 
thickness  of  the  rim  at  the  larger  end,  and  describe  the  circle 
limiting  it,  with  the  radius  v — i.  If  now  we  wrap  this 
development  back  on  the  cone,  we  can  mark  out  upon  its 
surface  the  outline  of  the  larger  ends  of  the  teeth ;  and  we 
observe  that  the  faces  of  the  teeth  will  thus  project  beyond 
the  pitch  cone,  and  the  flanks  lie  within  it.  If  we  next 
suppose  lines  to  be  drawn  from  the  vertex  Fto  the  several 
points  of  the  outline  of  the  teeth,  the  spaces  between  them, 
and  to  the  interior  circle  of  the  rim,  the  lines  so  drawn  will 
lie  on  the  bounding  surfaces  of  the  teeth  and  rim ;  and  the 
points  in  which  they  meet  the  surface  of  the  cone,  having  the 
vertex  v\  and  which  limits  the  smaller  ends  of  the  teeth,  will 
mark  out  on  this  surface  the  outlines  of  the  smaller  ends  of 


PROBLEMS   OF  MECHANISM.  179 

the  teeth   and  rim.     The   length  of  each  tooth,    measured 
along  the  element  V— n  of  the  pitch  cone,  will  be  n — ri. 

To  construct  the  vertical  projection  of  the  wheel,  we 
observe,  in  the  first  place,  that  the  points  n,  6,  e,  &c.  (Fig.  B), 
where  the  faces  and  flanks  join,  lie  upon  the  circumference 
of  the  circle  of  which  o — n  is  the  radius,  and  which  is  the 
pitch  circle  for  the  outline  of  the  ends  of  the  teeth ;  in  like 
manner  that  the  points  e,/  &c.,  of  the  apex  of  each  tooth,  lie 
on  a  circle  of  which  p — q  is  the  radius ;  the  points  s,  a,  rf, 
&c.,  lie  on  the  circle  of  which  r — s  is  the  radius ;  and  the 
interior  circle  of  the  rim  has  t — u  for  its  radius.  The  radii 
of  the  corresponding  circles,  on  the  smaller  ends  of  the  teeth 
and  rim,  are  o' — n';  p' — q';  r' — s';  and  t' — u'.  In  the  second 
place  all  these  circles  are  parallel  to  the  vertical  plane  of 
projection,  since  the  axis  of  the  wheel  is  perpendicular  to  this 
plane,  and  they  will  therefore  be  projected  on  this  plane  in 
their  true  dimensions. 

From  the  point  0  then,  the  vertical  projection  of  the  axis, 
describe  in  the  first  place  the  four  concentric  circles  with  the 
radii  0—Q,  0—N,  0—S,  and  0—  Z7  respectively  equal  to 
p — q,  o — n,  &c. ;  and,  from  the  same  centre,  the  four  others 
with  radii  0 — Q',  &C',  respectively  equal  to  p' — q',  &c. 

On  the  circle  having  the  radius  0 — N,  set  off  the  points 
N,  J3,  E,  &c.,  corresponding  to  n,  b,  e,  &c. ;  on  the  one  0 — Q, 
the  points  (7,  F,  &c.,  corresponding  to  c,  f,  &c. ;  on  the  one 
0 — S,  the  points  £,  A,  D,  &c.,  corresponding  to  5,  a,  c?,  &c. 
From  the  points  C  and  F,  thus  set  off,  draw  right  lines  to  the 
point  0;  the  portions  of  these  lines,  intercepted  between  the 
circles  of  which  0—Q  and  0 — Q'  are  respectively  the  radii, 
with  the  portions  of  the  arcs,  as  C— F,  O — F',  intercepted 
between  these  lines,  will  form  the  outline  of  the  vertical  pro- 
jection of  the  figure  of  the  apex  of  the  tooth.  The  portions 
of  the  lines,  drawn  from  B  and  E  to  0,  intercepted  between 
the  circles  described  with  the  radii  0 — JVand  0 — N',  together 
with  the  portions  of  the  lines  forming  the  edge  of  the  apex, 
and  the  curve  lines  B — C,  E—F  and  the  corresponding 
curves  B' — C",  F — E'  on  the  smaller  end,  will  be  the  projec 
tions  of  the  outlines  of  the  faces  and  flanks  of  the  tooth.  The 
outline  of  the  projection  of  the  bottom  of  the  space  will  lie 


ISO  INDUSTRIAL  DRAWING. 

bet-ween  the  right  lines  drawn  from  8  and  A  to  0,  and  the 
arcs  S- — A,  /S' — A',  intercepted  between  these  lines,  on  the 
circles  described  with  the  radii  0 — /S,  and  0 — S'. 

The  projection  of  the  cylindrical  eye  of  the  boss  is  the 
circle  described  with  the  radius  0 — K.  Having  completed 
the  vertical  projection,  the  corresponding  points  in  horizontal 
projection  are  found  by  projecting  the  points  C,  F,  B,  E,  At 
D,  &c.,  into  their  respective  circumferences  (Fig.  A)  at 
c',  f,  bf,  &c.  The  portions  of  the  lines  drawn  from  C  and  $ 
to  0,  in  vertical  projection,  will,  in  horizontal  projection,  be 
drawn  from  c'  and/'  to  V;  and  so  for  the  other  elements  of 
the  surfaces  of  the  faces,  flanks,  &c.,  of  the  teeth.  The  hori- 
zontal projection  of  the  larger  end  of  each  tooth  will  be  a 
figure  like  the  one  a'b'c'fe'd'. 

The  boss  projects  beyond  the  rim  at  the  larger  end  of  the 
tfheel ;  it  is  usually  a  hollow  cylinder.  Its  horizontal  pro- 
jection is  the  figure  xyzw,  &c. 

Prob.  117.  (PL  XY.  Fig.  140.)  To  construct  the  projec- 
tions of  the  same  wheel,  when  the  axis  is  oblique  to  the  vertical 
plane,  and  parallel  as  before  to  the  horizontal. 

This  variation  of  the  problem  requires  no  particular  verbal 
explanation;  as  from  preceding  problems  of  the  like  cha- 
racter, and  the  Figs,  the  manner  in  which  the  vertical 
projections  are  obtained  from  the  horizontal  will  be  readily 
made  out.  The  best  manner  however  of  commencing  the 
vertical  projection  will  be  to  draw,  in  the  first  place  (Fig.  Z>), 
all  the  ellipses  which  are  the  projections  of  the  circles 
described  with  the  radii  0 — Q,  0 — N,  &c.  (Fig.  C),  and  next 
those  of  the  vertices  of  the  three  cones,  which  will  be  the 
points  v,  0',  and  0".  These  being  drawn  the  projections  of 
the  different  lines,  forming  the  outline  of  the  projection  cf 
any  tooth,  can  be  readily  determined. 

Prob.  118.  (PI.  X.  Fig.  141.)  To  construct  the  projec- 
tions of  the  screw  with  a  square  thread. 

As  a  preliminary  to  this  problem,  it  will  be  requisite  to 
show  how  a  line  termed  a  helix,  can  be  so  marked  out  on  the 
surface  of  a  right  circular  cylinder  that,  when  this  surface  is 
developed  out,  the  helix  will  be  a  right  line  on  the  develop- 
ment ;  and  the  converse  of  this,  having  a  right  line  drawn  OD 


PROBLEMS   OF   MECHANiSM.  181 

the  developed  surface  of  a  right  circular  cylinder,  to  find  the 
projection  of  this  line,  when  the  development  is  wrapped 
around  the  surface. 

Let  ABCD  be  the  horizontal  projection  of  the  cylinder: 
acc'a'  its  vertical  projection ;  0  and  the  line  o — o'  the  projec- 
tions of  its  axis.  Let  the  circle  of  the  base  be  divided  into 
any  number  of  equal  parts,  for  example  eight,  and  draw  the 
vertical  projections  e — e',  f—f,  &c.,  corresponding  to  the 
points  of  division  E,  F,  &c.  Having  found  the  development 
of  this  cylinder,  by  constructing  a  rectangle  (PI.  X.  Fig. 
142),  of  which  the  base  a — a  is  equal  to  the  circumference  of 
the  cylinder's  base,  and  the  altitude  a— a'  is  that  of  the 
cylinder ;  through  the  points  e,  5,  /  c,  &c.,  respectively  equal 
to  the  equal  parts  A — E,  &c.,  of  the  circle,  draw  the  lines 
e — e',  b — &',  &c.,  parallel  to  a — a'.  These  lines  will  be  the 
developed  positions  of  the  elements  of  the  cylinder,  pro- 
jected in  e — efj  b — J',  &c.  Now  on  this  development  let  any 
inclined  line,  as  a — m,  be  drawn ;  and  from  the  point  n,  at 
the  same  height  above  the  point  a,  on  the  left,  as  the  point  m 
is  above  a  on  the  right,  let  a  second  inclined  line  n — m'  be 
drawn  parallel  to  a — m  ;  and  so  on  as  many  more  equidistant 
inclined  parallels  as  may  be  requisite.  Now  it  will  be 
observed,  that  the  first  line,  a — TO,  cuts  the  different  elements 
of  the  cylinder  at  the  points  marked  1,  2,  3,  &c. ;  and  there- 
fore when  the  development  of  the  cylinder  is  wrapped  around 
it  these  points  will  be  found  on  the  projections  of  the  same 
elements,  and  at  the  same  heights  above  the  projection  of  the 
base  as  they  are  on  the  development.  Taking,  for  example, 
the  elements  projected  in  b — &',  and  c — c',  the  points  2  and  4 
of  the  projection  of  the  helix  will  be  at  the  same  heights, 
b — 2,  and  c — 4  on  the  projections,  above  a — c,  as  they  are  on 
the  development  above  a — a.  It  will  be  further  observed, 
that  the  helix  of  which  a — m  is  the  development  will  extend 
entirely  around  the  cylinder ;  so  that  the  point  TO,  on  the 
projection,  will  coincide  with  the  two  m  and  n  on  the  deve- 
lopment, when  the  latter  is  wrapped  round ;  and  so  on  for 
the  other  points  w,  n',  and  m';  so  that  the  inclined  parallels 
will,  in  projection,  form  a  continuous  line  or  helix  uniformly 
wound  around  the  cylinder.  Moreover,  it  will  be  seen,  if 


182  INDUSTRIAL    DRAWING. 

through  the  points  1,  2,  3,  &c.,  on  the  development,  lines  are 
drawn  parallel  to  the  base  a — a,  that  these  lines  will  be  equi- 
distant, or  in  other  words  the  point  2  is  at  the  same  height 
above  1,  as  1  is  above  a,  &c. ;  and  that,  in  projection  also, 
these  points  will  be  at  the  same  heights  above  each  other; 
this  gives  an  easy  method  of  constructing  any  helix  on  a 
cylinde»  when  the  height  between  its  lowest  and  highest 
point  for  one  turn  around  the  cylinder  is  given.  To  show 
this;  having  divided  the  base  of  the  cylinder  into  any 
number  of  equal  parts  (PI.  X.  Fig.  141),  and  drawn  the 
vertical  projections  of  the  corresponding  elements,  set  off  from 
the  foot  of  any  element,  as  a,  at  which  the  helix  commences, 
the  height  a — ra,  at  which  the  helix  is  to  end  on  the  same 
element ;  divide  a — m  into  the  same  number  of  equal  parts 
as  the  base ;  through  the  points  of  division  draw  lines 
parallel  to  a — c,  the  projection  of  the  base ;  the  points  in 
which  these  parallels  cut  the  projections  of  the  elements  will 
be  the  required  points  of  the  projection  of  the  helix ;  draw- 
ing the  curved  line  a,  1,  2,  &c.,  through  these  points  it  will 
be  the  required  projection. 

Having  explained  the  method  for  obtaining  the  projec- 
tions of  a  helix  on  a  cylinder,  that  of  obtaining  the  projectiona 
of  the  parts  of  a  screw  with  a  square  fillet  will  be  easily 
understood. 

Prob.  119.  (PI.  XVI.  Fig.  143.)  To  construct  the  projections 
of  a  screw  with  a  square  fillet. 

Draw  as  before  a  circle,  with  any  assumed  radius  A — B, 
for  the  base  of  the  solid  cylinder  which  forms  what  is  termed 
the  newel  of  the  screw,  and  around  which  the  fillet  is  wrapped. 
Construct,  as  above,  the  projections  of  two  parallel  helices  on 
the  newel;  the  one  a2ra;  the  other  x2z;  their  distance 
apart,  a — x,  being  the  height,  or  thickness  of  the  fillet, 
estimated  along  the  element  a — a'  of  the  cylinder.  From  0, 
with  a  radius  0 — A',  describe  another  circle,  such  that 
A — A'  shall  be  the  breadth  of  the  fillet  as  estimated  in  a 
direction  perpendicular  to  the  axis  of  the  newel ;  and  let  the 
rectangle  a"c''d"e"  be  the  vertical  projection  of  this  cylinder. 
Having  divided  the  base  of  the  second  cylinder  into  a  like 
number  of  equal  parts  corresponding  to  the  first,  and  drawn 


PROBLEMS  OF  MECHANISM.  183 

the  vertical  projections  of  the  elements  corresponding  to  these 
points,  as  I — £',  &c.,  construct  the  vertical  projection  of  a 
helix  on  this  cylinder,  which,  commencing  at  the  point  a" 
shall  in  one  turn  reach  the  point  ra",  at  the  same  heighi 
above  a"  as  the  point  ra  is  above  a.  The  helix  thus  found 
will  evidently  cut  the  elements  of  the  outer  cylinder  at  the 
same  heights  above  the  base  as  the  corresponding  one  on  the 
inner  cylinder  cuts  the  corresponding  elements  to  those  of 
the  first ;  the  projections  of  the  two  will  evidently  cross  each 
other  at  the  point  2  on  the  line  b — b'.  In  like  manner 
construct  a  second  helix  x"22",  on  the  second  cylinder  and 
parallel  to  the  first,  commencing  at  a  point  x"  at  the  same 
height  above  a"  as  x  is  above  a.  This,  in  like  manner,  will 
cross  the  projection  x2z  at  the  point  2.  The  four  projections 
of  helices  thus  found  will  be  the  projections  of  the  exterior 
and  interior  lines  of  the  fillet ;  the  exterior  surface  of  which 
will  coincide  with  that  of  the  exterior  cylinder,  and  the  top 
and  bottom  surfaces  of  which  will  lie  between  the  correspond- 
ing helices  at  top  and  bottom.  The  void  space  between  the 
fillet  which  lies  between  the  exterior  cylinder  and  the  surface 
of  the  newel  is  termed  the  channel;  its  dimensions  are  usually 
the  same  as  those  of  the  fillet. 

Prob.  120.  (Pis.  XYI.  XVII.  Figs.  144  to  157.)  To 
construct  the  lines  showing  iJie  usual  combination  of  the 
working  beam,  the  crank,  and  the  connecting  rod  of  a  steam 
engine. 

In  a  drawing  of  the  kind  of  which  the  principal  object  is 
to  show  the  combination  of  the  parts,  no  other  detail  is  put 
down  but  what  is  requisite  to  give  an  idea  of  the  general 
forms  and  dimensions  of  the  main  pieces,  and  their  relative 
positions  as  determined  by  the  motions  of  which  they  are 
susceptible. 

As  each  element  of  this  combination  is  symmetrically  dis- 
posed with  respect  to  a  central  line,  or  axis,  we  commence 
the  drawing  by  setting  off,  in  the  first  place,  these  central 
lines  in  any  assumed  position  of  the  parts;  these  are  the 
lines  o— -f,  the  distance  from  the  centre  of  motion  of  the 
working  beam  A  to  that  of  its  connexion  with  the  connecting 
rod  B,  and  which  is  3  inches  and  55  hundredths  of  an  inch 


184  INDUSTRIAL  DRAWING. 

actual  measurement  on  the  drawing,  or  4  feet  43  hundredth? 
on  the  machine  itself,  the  scale  of  the  drawing  being  1  inch  tc 
l£  foot,  or  T'5  ;  next  the  line/— e,  the  distance  of  the  centre  of 
motion  /  to  that  e  of  the  connecting  rod  and  crank  C ;  lastly 
the  line  o — d,  from  the  centre  of  motion  e  to  that  d  of  the 
crank  and  the  working  shaft,  the  actual  distance  being  1  inch 
36  hundredths.  These  lines  being  accurately  set  off.  the 
outlines  of  the  parts  which  are  symmetrically  placed  with 
respect  to  them  may  be  then  set  off,  such  dimensions  as 
are  not  written  down  being  obtained  by  using  the  scale  of 
the  drawing,  or  from  the  more  detailed  Figs.  145  to  157. 

Having  completed  the  outlines  we  next  add  a  sufficient 
number  of  lines,  termed  indicating  lines,  to  show  the  ampli- 
tude of  motion  of  the  parts,  or  the  space  passed  over  between 
the  extreme  positions  of  the  axes,  as  well  as  the  direction  or 
paths  in  which  the  parts  move.  These  are  shown  by  the  arc 
described  from  o  with  the  radius  o—f;  the  circle  described 
with  d — e  ;  the  lines  o — a,  o — c,  and  o — 6,  the  extreme  and 
mean  positions  of  the  axis  o—f:  with  b — h,  and  b — g  the 
extreme  positions  off- — e. 

Besides  the  axes  and  indicating  lines,  others  which  may  be 
termed  axial  lines,  being  lines  drawn  across  the  centre  of 
motion  of  articulations,  as  through  the  point  of  the  axis  on 
cross  sections,  are  requisite,  for  the  full  understanding  of  the 
combinations  of  the  parts ;  such,  for  example,  as  the  lines 
z — x  and  v — w,  on  (Fig.  147),  which  is  a  cross  section  of  the 
connecting  rod,  made  at  m — n,  m' — ri,  Figs.  145,  146  ;  those 
X—  Y,  X'—Y,  &c.;  those  Z—W,  Z'—W  on  Figs.  148 
to  157. 

Prob.  121.  (PI.  XVIII.  Figs.  158  to  170.)  To  make  the 
measurements,  the  sketches,  and  finished  drawing  of  a  machine 
from  the  machine  itself. 

A  very  important  part  of  the  business  of  the  draftsman 
and  engineer  is  that  of  taking  the  measurements  of  industrial 
objects,  with  a  view  to  making  a  finished  drawing  from  the 
rough  sketches  made  at  the  time  of  the  measurements.  For 
the  purposes  of  this  labor,  the  draftsman  requires  the  usual 
instruments  for  measuring  distances  and  determining  the 
Horizontal  and  vertical  distances  apart  of  points ;  as  the 


PROBLEMS   OF   MECHANISM.  185 

carpenter's  rule,  measuring  rods,  or  tape,  compasses,  chalk 
line,  an  ordinary  level;  and  a  plumb  line.  The  first  three  are 
used  for  ascertaining  the  actual  distances  between  points, 
lines,  &c. ;  the  chalk  line  to  mark  out  on  the  parts  to  be 
measured  central  lines,  or  axes ;  the  two  last  to  determine 
the  horizontal  and  vertical  distances  between  points.  For 
sketching,  paper  ruled  into  small  squares  with  blue,  or  any 
other  colored  lines,  is  most  convenient ;  such  as  is  used,  for 
example,  by  engineers  in  plotting  sections  of  ground.  With 
such  paper,  or  lead  pencil,  and  pen  and  ink,  the  draftsman 
needs  nothing  more  to  note  down  the  relative  positions  of  the 
parts  with  considerable  accuracy.  Taking  for  example  the 
side  of  the  small  square  to  represent  one  or  more  units  of  the 
scale  adopted  for  the  sketch,  he  can  judge,  by  the  eye,  pretty 
accurately,  the  fractional  parts  to  be  set  off.  In  making 
measurements,  it  should  be  borne  in  mind,  that  it  is  better  to 
lose  the  time  of  making  a  dozen  useless  ones,  than  to  omit  a 
single  necessary  one.  The  sketch  is  usually  made  in  lead 
pencil,  but  it  should  be  put  in  ink,  by  going  over  the  pencil 
lines  with  a  pen,  as  soon  as  possible ;  otherwise  the  labor 
may  be  lost  from  the  effacing  of  numbers  or  lines  by  wear. 
The  lines  running  lengthwise  and  crosswise  on  the  paper, 
and  which  divide  its  surface  into  squares,  will  serve,  as 
vertical  and  horizontal  lines  on  the  sketch,  to  guide  the  hand 
and  eye  where  projections  are  required. 

It  is  important  to  remember,  that  in  making  measurements 
we  must  not  take  it  for  granted  that  lines  are  parallel  that 
seem  so  to  the  eye ;  as,  for  example,  in  the  sides  of  a  room, 
house,  &c.  In  all  such  cases  the  diagonals  should  be 
measured.  These  are  indispensable  lines  in  all  rectilineal 
figures  which  are  either  regular  or  irregular  except  the  square 
and  rectangle. 

The  mechanism  selected  for  illustrating  this  Prob.  is  the 
ordinary  machine  termed  a  crab  engine  for  raising  heavy 
weights.  It  consists,  1st  (Fig.  158),  of  a  frame  work  com- 
posed of  two  standards  of  cast  iron  A,  A,  connected  by 
wrought  iron  rods  J,  b  with  screws  and  nuts ;  the  frame  being 
firmly  fastened,  by  bolts  passing  through  holes  in  the  bottoms 
of  the  standards,  to  a  solid  bed  of  timber  framing ;  2d,  of  the 


186  INDUSTKIAL  DRAWING. 

mechanism  for  raising  the  weights,  a  drum  B  to  which  is 
fastened  a  toothed  wheel  C  that  gears  or  works  into  a  pinion 
D  placed  on  the  axle  a;  3d,  two  crank  arms  E  where  the 
animal  power  as  that  of  men  is  applied  ;  4th,  of  a  rope  wound 
round  the  drum,  at  the  end  of  which  the  resistance  or  weight 
to  be  raised  is  attached. 

The  sketch  (Figs.  159  to  168)  is  commenced  by  measuring 
the  end  view  A'  of  the  standards  and  other  parts  as  shown  in 
this  view ;  next  that  of  the  side  view,  as  shown  in  (Fig.  160). 
To  save  room,  the  middle  portion  of  the  drum  B",  &c.,  is 
omitted  here,  but  the  distances  apart  of  the  different  portions 
laid  down.  These  parts  should  be  placed  in  the  same 
relative  positions  on  the  sketch  as  they  will  have  in  projec- 
tion on  the  finished  drawing  (Figs.  169,  170).  The  drum 
being,  in  the  example  chosen,  of  cast  iron,  sections  of  a 
portion  of  it  are  given  in  Figs.  163,  164.  The  other  details 
speak  for  themselves. 

The  chief  point  in  making  measurements  is  a  judicious 
selection  of  a  sufficient  number  of  the  best  views,  and  then  a 
selection  of  the  best  lines  to  commence  with  from  which  the 
details  are  to  be  laid  in.  This  is  an  affair  of  practice.  The 
draftsman  will  frequently  find  it  well  to  use  the  chalk  line  to 
mark  out  some  guiding  lines  on  the  machine  to  be  copied, 
before  commencing  his  measurements,  so  as  to  obtain  central 
lines  of  beams,  &c. ;  and  the  sides  of  triangles  formed  by  the 
meeting  of  these  lines. 


TOPOGRAPHICAL  DRAWING.  187 


CHAPTER  XY. 

TOPOGRAPHICAL  DRAWING. 

THE  term  topographical  drawing  is  applied  to  the  methods 
adopted  for  representing  by  lines,  or  other  processes,  both 
the  natural  features  of  the  surface  of  any  given  locality,  and 
the  fixed  artificial  objects  which  may  be  found  on  the 
surface. 

This  is  effected,  by  the  means  of  projections,  and  profiles, 
or  sections,  as  in  the  representation  of  other  bodies,  com- 
bined with  certain  conventional  signs  to  designate  more 
clearly  either  the  forms,  or  the  character  of  the  objects  of 
which  the  projections  are  given. 

As  it  would  be  very  difficult  and,  indeed  with  very  few- 
exceptions,  impossible  to  represent,  by  the  ordinary  modea 
of  projection,  the  natural  features  of  a  locality  of  any  con- 
siderable extent,  both  on  account  of  the  irregularities  of  the 
surface,  and  the  smallness  of  the  scale  to  which  drawings  of 
objects  of  considerable  size  must  necessarily  be  limited,  a 
method  has  been  resorted  to  by  which  the  horizontal  dis- 
tances apart  of  the  various  points  of  the  surface  can  be  laid 
down  with  great  accuracy  even  to  very  small  scales,  and  also 
the  vertical  distances  be  expressed  with  equal  accuracy  either 
upon  the  plan,  or  by  profiles. 

To  explain  these  methods  by  a  familiar  example  which 
any  one  can  readily  illustrate  practically,  let  us  suppose 
(PL  XIX.  Fig.  171)  a  large  and  somewhat  irregularly  shaped 
potato,  melon,  or  other  like  object  selected,  and  after  being 
carefully  cut  through  its  centre  lengthwise,  so  that  the  section 
shall  coincide  as  nearly  as  practicable  with  a  plane  surface, 
let  one  half  of  it  be  cut  into  slices  of  equal  thickness  by 


188  INDUSTRIAL  DRAWING. 

sections  parallel  to  the  one  through  the  centre.  This  being 
done,  let  the  slices  be  accurately  placed  on  each  other,  so  as 
to  preserve  the  original  shape,  and  then  two  pieces  of  straight 
stiff  wire,  A,  B,  be  run  through  all  the  slices,  taking  care  to 
place  the  wires  as  nearly  perpendicular  as  practicable  to  the 
surface  of  the  board  on  which  the  bottom  slice  rests,  and  into 
which  they  must  be  firmly  inserted.  Having  marked  out 
carefully  on  the  surface  of  the  board  the  outline  of  the  figure 
of  the  under  side  of  the  bottom  slice,  take  up  the  slices,  being 
careful  not  to  derange  the  positions  of  the  wires,  and,  laying 
aside  the  bottom  slice,  place  the  one  next  above  it  on  the  two 
wires,  in  the  position  it  had  before  being  taken  up,  and, 
bringing  its  under  side  in  contact  with  the  board,  mark  out 
also  its  outline  as  in  the  first  slice.  The  second  slice  being 
laid  aside,  proceed  in  the  same  manner  to  mark  out  the  out- 
line on  the  board  of  each  slice  in  its  order  from  the  bottom ; 
by  which  means  supposing  the  number  of  slices  to  have  been 
five  a  figure  represented  by  Fig.  171  will  be  obtained.  Now 
the  curve  first  traced  may  be  regarded  as  the  outline  of  the 
base  of  the  solid,  on  a  horizontal  plane ;  whilst  the  other 
curves  in  succession,  from  the  manner  in  which  they  have 
been  traced,  may  be  regarded  as  the  horizontal  projections 
of  the  different  curves  that  bound  the  lower  surfaces  of  the 
different  slices ;  but,  as  these  surfaces  are  all  parallel  to  the 
plane  of  the  base,  the  curves  themselves  will  be  the  hori- 
zontal curves  traced  upon  the  surface  of  the  solid  at  the  same 
vertical  height  above  each  other.  With  the  projections  of 
these  curves  therefore,  and  knowing  their  respective  heights 
above  the  base,  we  are  furnished  with  the  means  of  forming 
some  idea  of  the  shape  and  dimensions  of  the  surface  in 
question.  Finally,  if  to  this  projection  of  the  horizontal 
curves  we  join  one  or  more  profiles,  by  vertical  planes  inter- 
secting the  surface  lengthwise  and  crosswise,  we  shall  obtain 
as  complete  an  idea  of  the  surface  as  can  be  furnished  of  an 
object  of  this  character  which  cannot  be  classed  under  any 
regular  geometrical  law. 

The  projections  of  the  horizontal  curves  being  given  as 
well  as  the  uniform  vertical  distance  between  them,  it  will  be 
very  easy  to  construct  a  profile  of  the  surface  by  any  vertical 


TOPOGRAPHICAL   DRAWING.  189 

plane.  Let  X—  T  be  the  trace  of  any  such  vertical  plane, 
and  the  points  marked  a,  a/,  x",  &c.,  be  those  in  which  it  cuts 
the  projections  of  the  curves  from  the  base  upwards.  Let 
G— L  be  a  ground  line,  parallel  to  X—  Y,  above  which  the 
points  horizontally  projected  in  x.  x',  x",  &c.,  are  to  be 
vertically  projected.  Drawing  perpendiculars  from  these 
points  to  G — L,  the  point  x  will  be  projected  into  the  ground 
line  at  y ;  that  marked  x'  above  the  ground  line  at  y\  at  the 
height  of  the  first  curve  next  to  the  base  above  the  horizontal 
plane  ;  the  one  marked  x",  will  be  vertically  projected  in  y", 
at  the  same  vertical  height  above  x1  as  y'  is  above  y;  and 
so  on  for  the  other  points.  "We  see  therefore  that  if  through 
the  points  y,  y',  y",  &c.,  we  draw  lines  y'— y',  &c.,  parallel  to 
the  ground  line  these  lines  will  be  at  equal  distances  apart, 
and  are  the  vertical  projections  of  the  lines  in  which  the 
profile  plane  cuts  the  different  horizontal  planes  that  contain 
the  curves  of  the  surface,  and  that  the  curve  traced  through 
yy'y",  &c.,  is  the  one  cut  from  the  surface.  In  like  manner 
any  number  of  profiles  that  might  be  deemed  requisite  to 
give  a  complete  idea  of  the  surface  could  be  constructed. 

In  examining  the  profile  in  connexion  with  the  horizontal 
projection  of  the  curves  it  will  be  seen  that  the  curve  of  the 
profile  is  more  or  less  steep  in  proportion  as  the  horizontal 
projections  of  the  curves  are  the  nearer  to  or  farther  from 
each  other.  This  fact  then  enables  us  to  form  a  very  good 
idea  of  the  form  of  the  surface  from  the  horizontal  projections 
alone  of  its  curves ;  as  the  distance  apart  of  the  curves  will 
indicate  the  greater  or  less  declivity  of  the  surface,  and  their 
form  as  evidently  shows  where  the  surface  would  present  a 
convex,  or  concave  appearance  to  the  eye. 

For  any  small  object,  like  the  one  which  has  served  for  our 
illustration,  the  same  scale  may  be  used  for  both  the  hori- 
zontal and  vertical  projections.  But  in  the  delineation  of 
large  objects,  which  require  to  be  drawn  on  a  small  scale,  to 
accommodate  the  drawing  to  the  usual  dimensions  of  the 
paper  used  for  the  purpose,  it  often  becomes  impracticable  to 
make  the  profile  on  the  same  scale  as  the  plan,  owing  to  the 
smallness  of  the  vertical  dimensions  as  compared  with  the 
horizontal  ones.  For  example,  let  us  suppose  a  hill  of  irre 


190         f  INDUSTRIAL  DRAWING. 

gular  shape,  like  the  object  of  our  preceding  illustration,  and 
that  the  horizontal  curve  of  its  base  is  three  miles  in  its 
longest  direction,  and  two  in  its  narrowest,  and  that  the 
highest  point  of  the  hill  above  its  base  is  ninety  feet ;  and  let 
us  further  suppose  that  we  have  the  projections  of  the  hori- 
zontal curves  of  the  hill  for  every  three  feet  estimated  ver- 
tically. Now  supposing  the  drawing  of  the  plan  made  to  a 
scale  of  one  foot  to  one  mile,  the  curve  of  the  base  would 
require  for  its  delineation  a  sheet  of  paper  at  least  3  feet  long 
and  2  feet  broad.  Supposing  moreover  the  projection  of  the 
summit  of  the  hill  to  be  near  the  centre  of  the  base  and  the 
declivity  from  this  point  in  all  directions  sensibly  uniform,  it 
will  be  readily  seen  that  the  distance  apart  of  the  horizontal 
curves,  estimated  along  the  longest  diameter  of  the  curve  of 
the  base  will  be  about  half  an  inch,  and  along  the  shortest 
one  about  one-third  of  an  inch ;  so  that  although  the  linear 
dimensions  of  the  horizontal  projections  are  only  the  y^W  of 
the  actual  dimensions  of  the  hill  yet  no  difficulty  will  be 
found  in  putting  in  the  horizontal  curves.  But  if  it  were 
required  to  make  a  profile  on  the  same  scale  we  should  at 
once  see  that  with  our  ordinary  instruments  it  would  be 
impracticable.  For  as  any  linear  space  on  the  drawing  is 
only  the  j^V  o-  part  of  the  corresponding  space  of  the  object,  it 
follows  that  for  a  vertical  height  of  three  feet,  the  distance 
between  the  horizontal  curves,  will  be  represented  on  the 
drawing  of  the  profile  by  the  T,V «  Part  of  a  f°ot>  a  distance 
too  small  to  be  laid  off  by  our  usual  means.  Now  to  meet 
this  kind  of  difficulty,  the  method  has  been  devised  of 
drawing  profiles  by  maintaining  the  same  horizontal  distances 
between  the  points  as  on  the  plan,  but  making  the  vertical 
distances  on  a  scale,  any  multiple  whatever  greater  than  that 
of  the  plan,  which  may  be  found  convenient.  For  example, 
in  the  case  before  us,  by  preserving  the  same  scale  as  that  of 
the  plan  for  the  horizontal  distances,  the  total  length  of  the 
profile  would  be  3  feet;  but  if  we  adopt  for  the  vertical 
distances  a  scale  of  Tl,7  of  an  inch  to  one  foot,  then  the  vertical 
distance  between  the  horizontal  curves  would  be  T35  of  an 
inch,  and  the  summit  of  the  profile  would  be  9  inches  above 
its  base.  It  will  be  readily  seen  that  this  method  will  not 


TOPOGEAPHICAL  DRAWING.  191 

alter  the  relative  vertical  distances  of  the  points  from  each 
other ;  for  T\  of  an  inch,  the  distance  between  any  two  hc~i 
zontal  curves  on  the  profile,  is  the  ^  of  9  inches  the  height 
of  the  projection  of  the  summit,  just  as  3  feet  is  the  ^  of  90 
feet  on  the  actual  object.  But  it  will  be  further  seen  that  cne 
profile  otherwise  gives  us  no  assistance  in  forming  an  idea  of 
the  actual  shape  and  slopes  of  the  object,  and  in  fact  rather 
gives  a  very  erroneous  and  distorted  view  of  them. 

Plane  of  Comparison,  or  Reference.  To  obviate  the  trouble 
of  making  profiles,  and  particularly  when  the  scale  of  the 
plan  is  so  small  that  a  distorted  and  therefore  erroneous  view 
may  be  given  by  the  profile  made  on  a  larger  scale  than  that 
of  the  plan,  recourse  is*  had  to  the  projections  alone  of  the 
horizontal  curves,  and  to  numbers  written  upon  them  which 
express  their  respective  heights  above  some  assumed  hori- 
zontal plane,  which  is  termed  the  plane  of  reference,  or  of 
comparison.  In  Fig.  171,  for  example,  the  plane  of  the  baae 
may  be  regarded  as  the  one  from  which  the  heights  of  all 
objects  above  it  are  estimated.  If  the  scale  of  this  drawing 
was  i  of  an  inch  to  %  an  inch,  and  the  actual  distance 
between  the  planes  of  the  horizontal  curves  was  equal  to  i 
an  inch,  then  the  curves  would,  in  their  order  from  the 
bottom,  be  h  an  inch  vertically  above  each  other.  To 
express  this  fact  by  numbers,  let  there  be  written  upon  the 
projection  of  the  curve  of  the  base  the  cypher  (0) ;  upon  the 
next  this  (1);  &c.  These  numbers  thus  written  will  indicate 
that  the  height  of  each  curve  in  its  order  above  that  of  the 
base  is  £,  f ,  f ,  &c.,  of  an  inch.  The  unit  of  measure  of  the 
object  in  this  case  being  half  an  inch.  The  numbers  so 
written  are  termed  the  references  of  the  curves,  as  they  indicate 
their  heights  above  the  plane  to  which  reference  is  made  in 
estimating  these  heights. 

The  selection  of  the  position  of  the  plane  of  comparison  is 
at  the  option  of  the  draftsman;  as  this  position,  however 
chosen,  will  in  no  respects  change  the  actual  heights  of  the 
points  with  respect  to  each  other  ;  making  only  the  references 
c/i  each  greater  or  smaller  as  the  plane  is  assumed  at  a  lower 
or  higher  level.  Some  fixed  and  well  defined  point  is  usually 
taken  for  the  position  of  this  plane.  In  the  topography  of 


192  INDUSTRIAL   DRAWING. 

loccilities  near  the  sea,  or  where  the  height  of  any  point  of  the 
locality  above  the  lowest  level  of  tide  water  is  known,  thig 
level  is  usually  taken  as  that  of  the  plane  of  comparison. 
This  presents  a  convenient  starting  point  when  all  the  curves 
of  the  surface  that  require  to  be  found  lie  in  planes  above  this 
level.  But  if  there  are  some  below  it,  as  those  of  the  exten- 
sion of  the  shores  below  low  water,  then  it  presents  a 
difficulty,  as  these  last  curves  would  require  a  different  mode 
of  reference  from  the  first  to  distinguish  them.  This  difficulty 
may  be  gotten  over  by  numbering  them  thus  (-1),  (-2),  &c., 
with  the  -  sign  before  each,  to  indicate  references  belonging 
to  points  below  the  plane  of  comparison.  The  better  method, 
however,  in  such  a  case,  is  to  assume*  the  plane  of  comparison 
at  any  convenient  number  of  units  below  the  lowest  water 
level,  so  that  the  references  may  all  be  written  with  numbers 
of  the  same  kind. 

References.  In  all  cases,  to  avoid  ambiguity  and  to  provide 
for  references  expressed  in  fractional  parts  of  the  unit,  the 
references  of  whole  numbers  alone  are  written  thus  (2.0),  that 
is  the  integer  followed  by  a  decimal  point,  and  a  0;  those  of 
mixed  or  broken  numbers,  thus  (2.30),  (3.58),  (0.37),  &c., 
that  is  with  the  whole  number  followed  by  two  decimal 
places  to  express  the  fractional  part. 

Projections  of  the  Horizontal  Curves.  No  invariable  rule 
can  be  laid  down  with  respect  to  the  vertical  distance  apart 
at  which  the  horizontal  curves  should  be  taken.  This 
distance  must  be  dependent  on  the  scale  of  the  drawing,  and 
the  purpose  which  the  drawing  is  intended  to  subserve.  In 
drawings  on  a  large  scale,  such  for  example  as  are  to  serve 
for  calculating  excavations  and  embankments,  horizontal 
curves  may  be  put  in  at  distances  of  a  foot,  or  even  at  less 
distances  apart.  In  maps  on  a  smaller  scale  they  may  be 
from  a  yard  upwards  apart.  Taking  the  scale  No.  8,  in  the 
Table  of  Scales  farther  on,  which  is  one  inch  to  50  feet,  or 
y£ 7,  as  that  of  a  detailed  drawing,  the  horizontal  curves  may 
be  put  in  even  as  close  as  one  foot  apart  vertically.  A  con- 
venient rule  may  be  adopted  as  a  guide  in  such  cases,  which 
is  to  divide  600  by  the  fraction  representing  the  ratio  which 
designates  the  scale,  and  to  take  the  resulting  quotient  to 


TOPOGRAPHICAL   DRAWING.  193 

express  the  number  of  feet  vertically  between  the  horizontal 
curves.  Thus  600  -4-  TJT  gives  one  foot  as  the  required 
distance;  600  ~  T?V«  gives  3  feet;  600  -j-  ^  gives  the 
half  of  a  foot,  &c.,  &c. 

But  whatever  may  be  this  assumed  distance  the  portion  of 
the  surface  lying  between  any  two  adjacent  curves  ia 
supposed  to  be  such,  that  a  line  drawn  from  a  point  on  the 
upper  curve,  in  a  direction  perpendicular  to  it  and  prolonged 
to  meet  the  lower,  is  assumed  to  coincide  with  the  real 
surface.  This  hypothesis,  although  not  always  strictly  in 
accordance  with  the  facts,  approximates  near  enough  to 
accuracy  for  all  practical  purposes;  especially  in  drawings 
made  to  a  small  scale,  or  in  those  on  a  large  one  where 
the  curves  are  taken  one  foot  apart  or  nearer  to  each 
other. 

Let  d  (Fig.  172)  for  instance  be  a  point  on  the  curve  (3.0), 
drawing  from  it  a  right  line  perpendicular  to  the  direction  of 
the  tangent  to  the  curve  (3.0)  at  the  point  d,  and  prolonging 
it  to  c  on  the  curve  (2.0),  the  line  d — c  is  regarded  as  the 
projection  of  the  line  of  the  surface  between  the  points  pro- 
jected in  d  and  c.  In  like  manner  a — b  may  be  regarded  as 
the  projection  of  a  line  on  the  portion  of  the  surface  between 
the  same  curves.  It  will  be  observed  however  that  the  line 
a — b  is  quite  oblique  with  respect  to  the  curve  (2.0),  whereas 
d—c  is  nearly  perpendicular  to  (2.0)  as  well  as  to  (3.0),  owing 
to  the  portions  of  the  curves  where  these  lines  are  drawn 
being  more  nearly  parallel  to  each  other  in  the  one  case  than 
in  the  other.  This  would  give  for  the  portion  to  which  a — b 
belongs  a  less  approximation  to  accuracy  than  in  the  other 
portion  referred  to.  To  obtain  a  nearer  degree  of  approxi- 
mation in  such  cases,  portions  of  intermediate  horizontal 
curves  as  x — x,  y — y,  &c.,  may  be  put  in  as  follows.  Suppose 
one  of  the  new  curves  y — y  is  to  be  midway  between  (2.0) 
and  (3.0).  Having  drawn  several  lines  as  a — b,  bisect  each 
of  them,  and  through  the  points  thus  obtained  draw  the 
curve  y — y,  which  will  be  the  one  midway  required.  In  like 
manner  other  intermediate  curves  as  x — cc,  y — y  may  be 
drawn.  Having  put  in  these  curves,  the  true  line  of 
declivity,  between  the  points  e  and  /  for  example,  will  b« 
13 


194  INDUSTRIAL   DRAWING. 

the  curved  or  broken  line,  e—f  cutting  the  intermediate 
curves  at  right  angles  to"  the  tangents  where  it  crosses 
them. 

The  intermediate  curves  are  usually  only  marked  In 
pencil,  as  they  serve  simply  to  give  the  position  of  the  line 
that  shows  the  direction  of  greatest  declivity  of  the  surface 
between  the  two  given  curves. 

Prob.  122.  Having  the  references  of  a  number  of  points  on  a 
drawing,  as  determined  by  an  instrumental  survey,  to  construct 
from  these  data  the  approximate  projection  of  the  equidistant 
horizontal  curves  having  whole  numbers  for  references. 

Engineers  employ  various  methods  for  determining  equi- 
distant horizontal  curves,  either  directly  by  an  instrumental 
process  on  the  ground,  or  by  constructions,  based  upon  the 
considerations  just  explained,  from  data  obtained  by  the 
ordinary  means  of  leveling,  &c. 

Let  us  suppose  for  example  that  ABCD  (Fig.  173)  repre- 
sents the'  outline  of  a  portion  of  ground  which  has  been 
divided  up  into  squares  of  50  feet  by  the  lines  x — x,  x' — x', 
y — y,  &c.,  run  parallel  to  the  sides  A — B  and  A — (7,  and 
that  pickets  having  been  driven  at  the  points  where  these 
lines  cut  each  other  and  the  parallel  sides,  it  has  been  deter- 
mined by  the  usual  methods  of  leveling  that  these  points 
have  the  references  respectively  written  near  them.  With 
these  data  it  is  required  to  determine  the  projections  of  the 
equidistant  horizontal  curves  with  whole  number  references 
which  lie  one  foot  apart  vertically. 

Having  set  off  a  line  A — x  (Fig.  174),  equal  to  A — cc,  on 
(Fig.  173)  draw  perpendiculars  to  it  at  the  points  A  and  x. 
From  these  two  points  set  off  along  the  perpendiculars  any 
number  of  an  assumed  unit  (say  half  an  inch  as  the  one 
taken),  and  divide  each  one  into  ten  equal  parts.  Through 
these  points  of  division  draw  lines  parallel  to  A — x. 

Cut  from  a  piece  of  stiff  paper  a  narrow  strip  like  A — G 
(Fig.  174),  making  the  edge  A — 0  accurately  straight.  By 
means  of  a  large  pin  fasten  this  strip  to  the  paper  and  draw- 
ing board  at  the  point  A. 

If  we  consider  that  for  the  distance  of  50  feet  between  any 
two  points  on  ground,  of  which  the  surface  is  uniform  (as  is 


TOPOGRAPHICAL  DRAWING.  195 

most  generally  the  case),  the  line  of  the  surface  between  the 
two  points  will  not  vary  very  materially  from  a  right  line, 
and  that  any  inconsiderable  difference  will  be  still  less 
sensible  on  a  drawing  of  the  usual  proportions,  we  m;iy 
without  any  important  error  then  assume  the  line  in  question 
to  be  a  right  line.  Now  as  the  reference  of  the  point  x  is 
(26.20)  the  difference  of  level  between  it  and  A,  or  the  height 
of  x  above  A  is  1.70  ft.,  or  equal  to  the  difference  of  the  two 
references.  But  from  what  has  just  been  laid  down  with 
respect  to  the  line  joining  the  points  A  and  x  drawn  on  the 
actual  surface,  it  is  plain  that  the  point  on  this  line  having 
the  whole  reference  (25.0)  lies  between  A  and  x,  and  that  as 
it  is  0.50  ft.  higher  than  A  its  projection  will  lie  between  A 
and  x  and  its  distance  from  A  will  be  to  the  distance  of  x 
from  A  in  the  same  proportion  as  its  height  above  A  is  to 
the  height  of  x  above  A,  or  as  0.50  ft.  is  to  1.70  ft.  By 
calculating,  or  by  constructing  by  (Prob.  54,  Fig.  55)  a  fourth 
proportional  to  A — x  =  50  ft.  ;  1.70  ft.  =  the  height  of  a 
above  A ;  and  0.50  ft.  =  the  height  of  the  required  point 
above  A;  we  shall  obtain  the  distance  of  the  projection  of 
this  point  from  A.  In  like  manner  by  calculation,  or 
construction,  we  can  obtain  the  distance  from  A  of  any 
other  point  between  A  and  x  of  which  the  reference  is 
given. 

But  as  the  calculation  of  these  fourth  proportionals  would 
require  some  labor  the  Fig.  174  is  used  to  construct  them  by 
this  simple  process.  Find  on  the  perpendicular  to  A — x  on 
the  right  the  division  point  marked  1.70 ;  turn  the  strip  of 
paper  around  its  joint  at  A  until  the  edge  A — 0  is  brought 
on  this  point,  and  confine  it  in  this  position.  The  portions 
of  the  parallels  intercepted  between  A — 0  and  the  perpen- 
diculars at  A  will  be  the  fourth  proportionals  required.  For 
example,  the  vertical  height  between  the  point  (24.50)  and 
the  one  (25.0)  being  equal  to  the  difference  of  the  two 
references,  or  0.50  foot,  the  horizontal  distance  which  corres- 
ponds to  this  is  at  once  obtained  by  taking  off  in  the  dividers 
the  distance,  on  the  parallel  drawn  through  the  point  .5, 
between  the  perpendiculars  at  A  and  the  edge  A — 0.  This 
distance  set  off  along  the  line  A— B  (Fig.  174)  from  A  to 


196  INDUSTRIAL   DRAWING. 

(25.0)  will  give  the  required  point.  In  li^e  manner  the 
distance  from  A  to  (26.0)  will  be  found,  by  taking  off  in  the 
dividers  the  portion  of  the  parallel  drawn  through  the  point 
1.5  on  the  perpendicular  at  A, 

To  find  the  points  corresponding  to  the  references  (24.0), 
(25.0),  and  (26.0),  on  the  line  x—x  parallel  to  A— D,  which 
lie  between  the  points  marked  (26.20)  and  (23.30),  the 
vertical  height  between  these  points  being  (26.20)  —  (23.30) 
=  2.90  feet,  first  bring  the  edge  A — 0  to  the  point  marked 
2.90  on  the  perpendicular  on  the  right,  then,  to  obtain  the 
distance  corresponding  to  (24.0),  take  off  the  portion  of  the 
intercepted  parallel  through  the  point  .7  and  set  it  off  from 
(23.30)  towards  (26.20),  and  so  on  for  the  other  points  (25.0) 
and  (26.0). 

Having  in  this  manner  obtained  all  the  points  on  the 
parallels  to  A — B  and  A — D,  with  entire  numbers  for 
references,  the  curves  drawn  through  the  points  having  the 
same  references  will  be  the  projections  of  the  corresponding 
horizontal  curves  of  the  surface. 

It  may  happen,  owing  to  an  abrupt  change  in  the  declivity 
of  the  ground  between  two  adjacent  angles  of  one  of  the 
squares,  as  at  b  between  the  point  A  and  y,  on  the  line 
A — D,  that  it  may  be  necessary  to  obtain  on  the  ground  the 
level  and  reference  of  this  point,  for  greater  accuracy  in 
delineating  the  horizontal  curves.  Suppose  the  reference  of 
b  thus  found  to  be  (21.50),  it  will  be  seen  that  the  rise  from 
y  to  b  is  only  0.4  foot,  whilst  from  b  to  A  it  is  3  feet.  To 
obtain  the  references  with  whole  numbers  between  b  and  A, 
take  off  the  distance  A— b  (Fig.  173)  and  set  it  off  from  A  to 
b  on  (Fig.  174),  and  through  b  erect  a  perpendicular  to  A — x, 
marking  the  point  where  this  perpendicular  cuts  the  parallel 
drawn  through  the  point  3,  and  bringing  the  edge  A — 0  of 
the  strip  of  paper  on  this  point,  we  can  obtain  as  before  the 
distances  to  be  set  off  from  b  towards  A  (Fig.  173)  to  obtain 
the  required  references. 


TOPOGRAPHICAL  DRAWING.  197 


CONVENTIONAL  METHODS  OF  REPRESENTING  THE  NATURAL 
AND  ARTIFICIAL  FEATURES  OF  A  LOCALITY. 

For  the  purposes  of  an  engineer,  or  for  the  information  of 
a  person  acquainted  with  the  method,  that  of  representing 
the  surface  of 'the  ground  by  the  projections  of  equidistant 
horizontal  curves  is  nearly  all  that  is  requisite ;  but  to  aid 
persons  in  general  to  distinguish  clearly  and  readily  the 
various  features  of  a  locality,  certain  conventional  means  are 
employed  to  express  natural  features  as  well  as  artificial 
objects,  which  are  termed  topographical  signs. 

Slopes  of  ground.  The  line  of  the  slope,  o  declivity  of  the 
surface  at  any  given  point  between  any  two  equidistant 
horizontal  curves,  it  has  been  shown  is  measured  along  a 
right  line  drawn  from  the  upper  to  the  lower  curve,  and 
perpendicular  to  the  tangent  to  the  upper  curve  at  the  given 
point.  This  slope  may  be  estimated  either  by  the  number 
of  degrees  in  the  angle  contained  between  the  line  of  declivity 
and  a  horizontal  line,  in  the  usual  way  of  measuring  such 
angles ;  or  it  may  be  expressed  by  the  ratio  between  the 
perpendicular  and  base  of  a  right  angle  triangle,  the  vertical 
distance  between  the  equidistant  horizontal  curves  being  the 
perpendicular,  and  the  projection  of  the  line  of  declivity  the 
base.  If  for  example  the  line  of  declivity  of  which  a — b 
(Fig.  172)  is  the  projection  makes  an  angle  of  45°  with  the 
horizontal  plane,  then  the  vertical  distance  between  the 
points  a  and  b  on  the  two  curves  will  be  equal  to  a — 6,  and 
the  ratio  between  the  perpendicular  and  base  of  the  right 
angle  triangle,  by  which  the  declivity  in  this  case  is  esti- 
mated is  —  =  —  since  the  equidistant  curves  are  taken  one 
ab  1 

unit  apart. 

As  a  general  rule  all  slopes  greater  than  45°  or  y  are 
regarded  as  too  precipitous  to  be  expressed  by  horizontal 
equidistant  curves,  the  most  that  is  done  to  represent  them 
is  to  draw  when  practicable  the  top  and  bottom  lines  of  the 
surface.  In  like  manner  all  slopes  less  than  0°..53'..43" 
or  ?V  are  regarded  as  if  the  surface  were  horizontal;  stil. 


198  INDUSTRIAL   DRAWING. 

upon  such  slopes  the  horizontal  curves  may  when  lequisite 
be  put  in ;  but  nothing  further  is  added  to  express  the 
declivity  of  the  surface. 

Lines  of  declivity,  &c.  The  lines  used  in  topographical 
drawing  to  picture  to  the  eye  the  undulations  of  the  ground, 
and  which  are  drawn  in  the  direction  of  the  lines  of  declivity 
of  the  surface,  serve  a  double  purpose,  that  of  a  popular 
representation  OT  the  object  expressed,  and  with  which  most 
intelligent  persons  are  conversant,  and  that  of  giving  the 
means,  when  they  are  drawn  in  accordance  to  some  system 
agreed  upon,  of  estimating  the  declivities  which  they  figure, 
with  all  the  accuracy  required  in  many  practical  purposes 
for  which  accurate  maps  are  consulted  by  the  engineer  or 
others. 

As  the  horizontal  curves  when  accompanied  by  their 
references  to  some  plane  of  comparison  are  of  themselves 
amply  sufficient  to  give  an  accurate  configuration  of  the 
surface  represented,  it  is  not  necessary  to  place  on  such 
drawings  the  lines  used  on  general  maps,  and  which  to  a 
certain  extent  replace  the  horizontal  curves.  The  lines  of 
declivity  in  question  will  therefore  be  confined  to  maps  on  a 
somewhat  small  scale,  in  which  horizontal  curves  are  not 
resorted  to  with  any  great  precision,  although  they  may  have 
been  used  to  some  extent  as  a  general  guide  in  constructing 
the  outlines  of  the  map,  such  for  example  as  from  one  inch 
to  100  feet,  or  r¥Vo>  and  upwards  as  far  as  such  lines  can 
serve  any  purpose  of  accuracy,  say  one  inch  to  half  a  mile,  or 

3"!  «•«•(-• 

To  represent  therefore  the  form  and  declivities  of  all 
slopes,  from  |  to  -6V  inclusive,  in  maps  on  these  and  inter- 
mediate scales,  the  following  rules  may  be  followed  for  pro 
portioning  the  breadth  and  the  length  of  the  lines  oJ 
declivity,  and  the  blank  spaces  between  them. 

1st,  The  distance  between  the  centre  lines  of  the  lines  of  declivity 
shall  be  2  hundredth*  of  an  inch  added  to  the  %  of  the  denomi- 
nator of  the  fraction  denoting  the  declivity  expressed  in  hun- 
dredths  of  an  inch. 

Thus  for  example  in  the  declivity  denoted  6-'T  the  rule 
•jpves  (2  4-  V )  =  18  hundredths  of  an  inch  for  the  distance 


TOPOGRAPHICAL   DRAWING.  199 

apart  of  the  lines.  In  the  declivity  of  £  we  obtain  (2  4-  I) 
=  2$  Imndredths  of  an  inch. 

2d.  The  lines  should  be  the  heavier  as  they  are  nearer  tc 
each  other,  or  as  the  declivity  expressed  by  them  is  the 
steeper.  For  the  most  gentle  slope  so  expressed,  that  of  Ty, 
the  lines  should  be  fine,  for  those  of  -J,  or  steeper,  their  breadth 
should  be  1$  hundredths  of  an  inch. 

This  rule  will  make  the  blank  space  between  the  heavy 
strokes  equal  to  half  the  breadth  of  the  stroke. 

3d.  No  absolute  rules  can  be  laid  down  with  respect  to  the 
lengths  of  the  strokes,  these  will  depend  upon  the  scale  of  the 
drawing,  the  skill  of  the  draftsman,  and  the  form  of  the  surface 
to  be  denned  by  them.  If  we  take  for  example  the  scale  of 
5-^0  or  one  inch  to  50  feet,  and  suppose  the  horizontal  curves 
to  be  put  in  at  one  foot  apart  vertically,  which  on  the  draw- 
ing corresponds  to  ^  or  2  hundredths  of  an  inch,  the  distance 
between  these  curves  on  slopes  of  -j-  would  be  2  hundredths 
of  an  inch,  whilst  on  a  slope  of  Jy,  the  curves  would  be 
2  x  64  =  128  hundredths,  or  1.28  in.,  nearly  an  inch  and 
one  third  apart.  In  the  first  case  therefore  if  the  strokes 
were  limited  between  the  two  curves  of  each  zone  they  would 
be  only  2  hundredths  of  an  inch  long,  whilst  in  the  second, 
if  a  like  limit  were  prescribed,  they  would  be  an  inch  and  a 
third  in  length  ;  both  of  which  would  be  inconvenient  to  the 
draftsman,  and  would  present  an  awkward  appearance,  par- 
ticularly the  latter,  on  the  drawing.  To  obviate  this  difficulty 
then  it  has  been  found  well,  on  gentle  slopes,  to  limit  the 
length  of  the  stroke  to  about  6  tenths  of  an  inch,  and  in 
steep  slopes  to  adopt  strokes  of  the  length  from  8  to  16  hun- 
dredths of  an  inch. 

These  limits  will  require  on  steep  slopes  to  certain  scales 
that  the  strokes  shall  embrace  the  zones  comprised  between 
three  or  more  horizontal  curves,  whilst  on  gentle  slopes  to 
some  scales  it  will  be  necessary  to  divide  up  the  zone  com- 
prised by  two  curves  into  two  or  more  by  intermediate  curves 
in  pencil,  so  as  to  obtain  auxiliary  zones  of  convenient 
breadth  for  the  draftsman  between  which  the  strokes  are  put 
in,  according  to  the  1st  and  2d  rules,  the  strokes  of  one 
auxiliary  zone  not  running  into  those  of  the  other. 


200  INDUSTRIAL  DRAWING. 

Practical  applications.  Suppose  on  a  zone  between  the 
curves  (2.0)  and  (3.0)  that  the  distance  between  the  points  f 
and /is  1.2  in.,  or  120  hundredths  of  an  inch,  it  would  be 
necessary  according  to  the  3d  rule  to  divide  this  zone  into  at 
least  two  by  one  auxiliary  curve.  Let  us  suppose  it  to  be 
divided  into  four  parts  by  three  auxiliary  curves  x — x,  y — yt 
z — z  put  in  according  to  what  has  been  already  laid  down, 
Having  done  this  calculate  by  rule  1st  the  distance  occupied 
along  this  curve  by  5  strokes  or  lines  of  declivity.  Supposing 
the  slope  to  be  Jv,  the  rule  would  give  (2  +  V)  =  17 
hundredths  of  an  inch  for  the  distance  apart  of  two  strokes  ; 
and  for  five  it  would  give  4  x  17  =  68  hundredths.  Take 
now  a  strip  of  paper  and  set  off  on  its  edge  68  hundredths  of 
an  inch,  which  divide  into  four  equal  parts ;  then  apply  this 
edge  to  the  curve  z — z  and  set  off  from  o  to  p  the  dots  for  the 
five  strokes ;  do  the  same  for  the  curves  y — y,  and  x — x,  and 
through  the  points  thus  set  off  draw  the  strokes  normal  to 
the  curve  along  which  they  are  set  off. 

Where  the  curves  approach  nearer  to  each  other,  and  are 
less  than  6  tenths  of  an  inch  apart  and  over  4  tenths,  as  at 
d- — c,  it  will  be  well  to  draw  an  intermediate  line  as  m — n 
along  which  the  strokes  will  be  set  off,  and  to  which  they 
will  be  drawn  perpendicularly. 

Lines  of  declivity  put  in  accurately  in  this  manner,  in 
groups  of  five,  from  distance  to  distance  between  the  hori- 
zontal curves,  will  serve  to  guide  the  hand,  in  judging  by 
the  eye  the  positions  of  the  intermediate  lines  between  the 
groups  ;  the  spaces  gradually  contracting,  or  widening,  as  the 
slope,  as  shown  by  the  positions  of  the  horizontal  curves, 
becomes  steeper,  or  more  gentle. 

Scale  of  spaces.  When  the  spaces  between  the  lines  of 
declivity  have  been  carefully  put  in  according  to  the  pre- 
ceding system,  they  will  serve  to  determine  the  declivity  at 
any  point ;  and  a  scale  of  spaces,  corresponding  to  the 
declivities,  ought  to  be  put  down  on  the  drawing,  in  like 
manner  as  we  put  down  a  scale  for  ascertaining  horizontal 
distances.  The  following  method  may  be  taken  to  construct 
this  scale.  On  a  right  line  estimating  from  the  point  A 
(PI.  XIX.  Fig.  175)  set  off  64  equal  parts  to  J5,  each  part 


TOPOGRAPHICAL    DRAWING.  £01 

being  equal  say  to  Jff,  or  TL  of  an  inch.  Number  the  points 
of  division  from  o  at  A,  to  64  at  R  Construct  perpendiculars 
to  the  right  line  at  A  and  B,  and  on  the  one  at  A  set  off  a 
distance  to  C  corresponding  to  four  spaces  of  the  lines  of 
declivity  for  the  slope  of  },  and  at  B  for  spaces  to  D  for  the 
slope  of  v\.  Draw  a  right  line  C—  D  through  the  points 
thus  set  off.  Through  each  of  the  equal  divisions  on  A — B, 
or  through  every  fifth  one,  draw  lines  parallel  to  the  two 
perpendiculars;  each  of  these  lines,  intercepted  between 
A — B  and  C—  D,  will  represent  four  spaces,  corresponding 
to  the  slope  marked  at  the  points  on  A — B. 

To  find  the  declivity  of  a  zone  between  two  horizontal 
curves,  at  any  point,  from  the  scale,  we  take  off  in  the 
dividers  the  distance  of  four  spaces  of  the  lines  of  declivity 
at  the  point,  then  place  the  points  of  the  dividers  on  the  lines 
A — B  and  C- — D  so  that  the  line  drawn  between  the  points 
will  be  perpendicular  to  A — B,  the  corresponding  number  on 
A — B  will  give  the  slope.  Suppose  for  example  the  points 
of  the  dividers  when  placed  embrace  the  points  ra  and  w, 
the  corresponding  number  on  A — B  being  about  34  gives  ^ 
for  the  required  slope. 

Surfaces  of  water.  (PI.  XX.  Fig.  176.)  To  represent  water 
a  series  of  wavy  lines  A,  A  are  drawn  parallel  to  the  shores. 
The  lines  near  the  shores  are  heavier  and  nearer  together 
than  those  towards  the  middle  of  the  surface.  No  definite 
rule  can  be  laid  down  further  than  to  make  the  lines  finer 
and  to  increase  the  distance  between  them  as  they  recede 
from  the  shore.  When  the  banks  are  steep  the  slope  is 
represented  by  heavy  lines  of  declivity.  The  water  line  is  a 
tolerably  heavy  line. 

If  islands  B  occur  in  the  water  course,  some  pains  must  be 
taken  in  uniting  the  water  lines  around  its  shores  with  the 
others. 

Shores.  Sandy  shelving  shores  C  are  represented  by  fine 
dots  uniformly  spread  over  the  part  they  occupy  on  the 
drawing.  The  dots  are  strewn  the  more  thickly  as  the  shore 
is  steeper. 

Gravelly  shores  are  represented  by  a  mixture  of  fine  and 
coarse  dots. 


202  INDUSTRIAL  DRAWING. 

Meadows.  These  are  represented  E  by  systems  of  very 
short  fine  lines  placed  in  fan  shape,  so  as  to  give  the  idea  of 
tufts  of  grass.  The  tufts  should  be  put  in  uniformly, 
parallel  to  the  lower  border  of  the  drawing,  so  as  to  produce 
a  uniform  tint. 

Marshy  ground.  This  feature  F,  F  is  represented  by  a 
combination  of  water  and  grass,  as  in  the  last  case.  The 
lines  for  the  water  surfaces  are  made  straight,  and  varied  in 
depth  of  tint,  giving  the  idea  of  still  water  with  reflections 
from  its  surface. 

Trees.  Single  trees  /,  /  are  represented  either  by  a  tuft 
resembling  the  foliage  of  a  bush,  with  its  shadow,  a  small 
circle,  or  a  black  dot,  according  to  the  scale  of  the  drawing. 
Evergreens  may  be  distinguished  from  other  trees  by  tufts 
of  fine  short  lines  disposed  in  star  shape.  Forests  O  are 
represented  by  a  collection  of  tufts,  small  circles,  and  points, 
so  disposed  as  to  cover  the  part  uniformly.  Brushwood  E 
and  clearings  with  undergrowth  standing,  with  smaller  and 
more  sparse  tufts,  &c.  Orchards  as  in  0. 

Rivulets,  ravines,  &c.  Small  water-courses  of  this  kind 
K,  K  and  their  banks  are  represented  by  the  shore  lines 
or  bank  slopes,  when  the  scale  of  the  drawing  is  large 
enough  to  give  the  breadth  of  the  stream.  The  lines  gra- 
dually diverging,  or  else  made  farther  apart  below  the 
junction  of  each  affluent.  On  small  scales  a  single  line 
is  used,  which  is  gradually  increased  in  heaviness  below 
each  affluent. 

Rocks  This  feature  L  is  expressed  by  lines  of  more  or 
less  irregularity  of  shape,  so  disposed  as  to  give  an  idea  of 
rocky  fragments  interspersed  over  the  surface,  and  connected 
by  lines  with  the  other  portions  intended  to  represent  the 
mass  of  whole  rock. 

Artificial  objects.  The  above  are  the  chief  natural  features 
represented  conventionally.  The  principal  conventional 
signs  for  artificial  objects  will  be  best  gathered  from  Plates 
XIX.  and  XX. 

In  most  works  for  elementary  instruction,  and  in  the 
systems  of  topographical  signs  adopted  in  public  services, 
almost  every  natural  and  artificial  feature  has  its  representa- 


TOPOGRAPHICAL  DRAWING.  203 

live  sign.  The  copying  of  these  is  good  practice  for  the 
pupil ;  but  for  actual  service  those  signs  alone  which  desig- 
nate objects  of  a  somewhat  permanent  character  are  strictly 
requisite;  as  in  culture,  for  example,  rice  fields  may  be 
expressed  by  a  sign,  as  they,  for  the  most  part,  retain  for  a 
long  time  this  destination  ;  whereas  the  ploughed  field  of  the 
Spring  is  in  grain  in  Summer  and  barren  in  Winter ;  and  the 
field  of  Indian  corn  of  this  year  is  in  wheat  the  next,  &c., 
&c. 

Practical  methods.  Finished  topographical  drawings  form 
a  part  of  the  office  work  of  the  civil  engineer,  that  require 
great  time,  skill,  and  care.  For  field  duties  he  is  obliged  to 
resort  to  methods  more  expeditious  in  their  results  than 
those  of  the  pen,  and  the  use  of  the  lead  pencil  furnishes  one 
of  the  best.  The  draftsman  should  therefore  accustom  him- 
self to  sketch  in  ground  by  the  eye,  and  endeavor  to  give  to 
his  sketch  at  once,  without  repeated  erasures  and  interlinea- 
tion, the  final  finish  that  it  should  receive  to  subserve  his 
purposes.  Hill  slopes,  horizontal  curves,  water,  &c.,  &c., 
may  be  sketched  in  either  by  lines,  according  to  rules 
already  laid  down,  or  else  by  uniform  tints  obtained  by 
rubbing  the  pencil  over  the  paper  until  a  tint  is  obtained  of 
such  intensity  as  to  represent  the  general  effect  of  lines  of 
declivity  of  varying  grade,  water  lines,  &c.  The  pencil  used 
for  this  purpose  should  be  very  black  and  moderately  hard, 
so  as  to  obtain  tints  of  any  depth,  from  deep  black  to  the 
lightest  shade  which  will  not  be  easily  effaced.  The  effects 
that  may  be  produced  in  this  manner  are  very  good,  and 
considerable  durability  may  be  given  to  the  drawing  by 
pasting  the  paper  on  a  coarse  cotton  cloth,  and  then  wetting 
the  surface  of  the  drawing  with  a  mixture  of  milk  and  water 
half  and  half.  Every  draftsman  will  do  well  to  exercise 
himself  at  this  work  in  the  office  until  he  finds  he  can 
imitate  any  given  ground  by  tints. 

Colored  Topography.  The  use  of  colors  in  topography  is 
an  effective  and  rapid  method  of  indicating  the  features  of 
land,  and  one  largely  employed. 

The  colors  used  are  indigo,  Hooker's  green,  No.  2,  yellow 
ochre,  burnt  sienna,  carmine,  gamboge,  and  sepia. 


204  INDUSTRIAL   DRAWING. 

Water  (a  PL  XXYIII.)  is  indicated  by  a  flat  tint  of  indigo  ; 
it  is  shaded  out  from  the  shore-line,  when  there  is  one. 

Grass-land  (b  PL  XXVIII.)  is  indicated  by  a  flat  tint  of 
green. 

Sand,  roads,  and  streets  (c  PL  XXYIII.)  are  indicated  by 
a  fiat  tint  of  yellow  ochre. 

JSuildings,  bridges,  and  all  structures  (d.  PL  XXYIII.), 
are  indicated  by  a  tint  of  carmine. 

Railroads  (e.  PL  XXYIII.)  are  indicated  by  a  dark  line 
of  carmine  without  cross  lines. 

Cultivated  land  (f.  PL  XXYIII.)  is  indicated  by  a  flat 
tint  of  burnt  sienna;  sometimes  parallel  lines  (g.  PL  XXYIII.) 
are  ruled  over  the  flat  tint,  using  for  the  purpose  either  a 
darker  tint  of  burnt  sienna,  green,  or,  more  commonly,  sepia. 

Uncultivated  land  (h.  PL  XXYIII.)  is  indicated  by  a 
double  tint  of  burnt  sienna  arid  green.  To  lay  a  double  tint, 
prepare  the  two  tints  in  separate  saucers,  then  using  a  brush 
for  each  tint,  carry  one  color  for  a  short  distance  upon  the 
surface,  and  then  change  for  the  other  color  and  brush,  letting 
the  colors  join  and  blend  of  themselves ;  alternate  the  tints 
in  this  way  until  the  whole  surface  is  covered.  Avoid  any 
regularity  in  the  mottled  tint  obtained. 

Hitts  (i.  PL  XXYIII.)  are  indicated  by  a  tint  of  sepia,  the 
depth  of  the  tint  corresponding  to  the  slope  of  the  land,  being 
darkest  where  the  slope  is  greatest,  and  becoming  lighter  as 
the  slope  decreases.  The  sepia  is  laid  over  the  land  tints. 

Trees  (L  PL  XXYIII.).  There  are  a  number  of  steps  to 
be  followed  in  indicating  trees,  l&t,  lay  the  land  tint ;  2d, 
pencil  in  fine  lines  the  outlines  of  the  trees  ;  3d,  tint  them  with 
green,  making  the  lower  right-band  part  the  darkest ;  4th, 
touch  up  the  trees  with  gamboge  upon  the  light  side  (the 
upper  left-hand) ;  5th,  add  the  shadows  of  the  trees  with  sepia. 

Marshy  ground  (1.  P) .  XXYIII.)  is  indicated  by  water  and 
grass  land  so  arranged  that  the  position  of  the  patches  of  land 
shall  be  horizontal;  draw  a  shade  line  of  sepia  along  the 
lower  edges  of  the  Jand. 

Light  is  supposed  to  come  from  the  upper  left-hand  corner 
of  the  drawing ;  hills  are  shaded  without  any  reference  to  the 
direction  of  light ;  only  the  shadows  of  such  objects  as  houses, 
trees,  etc.,  are  represented  ;  sepia  is  used  for  shadows. 


TOPOGRAPHICAL  DRAWING.  205 

For  the  method  of  preparing  and  using  tints,  see  the  chap- 
ter on  tinting. 

When  making  a  colored  plate,  first  pencil  everything,  then  lay 
the  flat  tints,  then  touch  up  the  trees, then  the  hills  and  shadows. 

The  division  lines  between  fields  and  all  outlines  are  ruled 
with  sepia. 

For  pen  drawings  the  draftsman  should  always  have  at 
hand  a  good  supply  of  pens  made  of  the  best  quills,  with  nebs 
of  various  sizes  to  suit  lines  of  various  grades,  for  slopes,  &c. 
His  ink  should  be  of  the  best,  and  of  a  decided  tint  when  laid 
on  ;  deep  black,  red,  green,  &c. 

The  breadth  of  lines  adopted  for  different  objects  must 
depend  upon  the  importance  of  the  object,  and  the  magnitude 
of  the  scale  to  which  the  drawing  is  made.  In  drawings  to 
small  scales  lines  of  not  more  than  two  breadths  can  be  used, 
as  the  fine  and  medium.  For  those  to  larger  scales,  three 
sizes  of  lines  may  be  introduced,  the  fine,  medium,  and 
heavy. 

Similar  remarks  may  be  made  on  lettering  and  the  size, 
&c.,  of  borders.  To  letter  well  requires  much  practice  from 
good  models.  The  draftsman  should  be  able  to  sketch  in  by 
the  eye  letters  of  every  character  and  size  without  resorting 
to  rulers  or  dividers  ;  until  he  can  do  this,  whatever  pains  he 
may  take,  his  lettering  will  be  stiff  and  ungainly.  The  size 
of  the  lettering  will  be  dependent  upon  that  of  the  drawing 
and  the  importance  of  the  object.  The  character  is  an  affair 
of  good  taste,  and  is  best  left  to  the  skill  and  fancy  of  the 
draftsman  ;  for  arbitrary  rules  cannot  alone  suffice,  even  were 
they  ever  rigorously  attended  to. 

As  it  is  of  some  importance  to  obtain  the  best  effects  in 
drawings  which  demand  so  much  time  and  labor  as  topo- 
graphical maps,  it  may  be  well  to  observe  that,  in  pen  or 
line  drawings,  it  is  best  to  put  in  the  letters  before  the  lines 
of  declivity,  water  lines,  &c. ;  as  it  is  less  difficult  to  put  in 
the  lines  without  disfiguring  the  letters  than  to  make  clean 
and  well  defined  letters  over  the  lines. 

Tne  border  of  the  drawing,  like  the  lettering,  is  frequently 
a  fancy  composition  of  the  draftsman.  It  most  generally 
consists  of  a  light  line  on  the  interior  and  a  heavy  one  on  the 


206  INDUSTRIAL    DUAWINO. 

exterior ;  the  heavy  line  having  the  same  breadth  as  that  of 
the  blank  space  between  it  and  the  light  line.  As  the  bordei 
is  generally  a  rectangle  in  shape,  the  rule  usually  followed 
for  proportioning  its  breadth — which  includes  the  light  line, 
the  blank  space,  an<3  the  heavy  line — is  to  make  it  the  one 
hundredth  part  of  the  length  of  the  shorter  side  of  the  rect- 
angle. 

The  title  of  the  drawing  is  placed  without  the  border  at 
top  when  it  takes  up  but  one  line ;  when  it  requires  several 
it  is  usually  placed  within  it.  The  greatest  height  of  the 
letters  of  the  title  should  be  three  hundredths  of  the  length 
of  the  shorter  side  of  the  border ;  and  when  the  title  is  with- 
out the  border  the  blank  space  between  it  and  the  border 
should  be  from  two  to  four  hundredths  of  the  shorter 
side. 

To  every  line  of  topographical  drawing  there  should  be 
two  scales,  one  to  express  the  horizontal  distances  between 
the  points  laid  down ;  the  other  a  scale  to  express  the  slopes 
as  in  Fig.  175.  The  scales  should  be  at  the  bottom  of  the 
drawing,  either  within  or  without  the  border,  according  to 
the  space  unoccupied  by  the  drawing. 

Finally  every  drawing  should  receive  the  signature  of  the 
draftsman;  the  date  of  the  drawing;  and  state  from  what 
authorities,  or  sources  compiled ;  and  under  whose  direction, 
or  supervision  executed.  If  emanating  from  any  recognized 
public  office,  it  ought  also  to  be  stamped  with  the  seal  of  the 
office. 

Scales  of  distances.  In  our  corps  of  military  engineers,  for 
the  purposes  of  preserving  uniformity  and  attaining  accuracy 
in  the  execution  of  maps  and  plans  for  official  action,  a 
system  of  regulations  is  adopted,  to  the  requirements  of 
which  strict  conformity  is  enjoined  on  all  in  any  way 
connected  with  those  corps,  prescribing  the  manner  in  which 
all  objects  are  to  be  represented,  and  the  scales  to  which  the 
drawings  of  them  shall  be  made.  As  the  last  point  is  the 
result  of  much  experience,  and  may  save  the  young  drafts- 
man much  time  in  the  selection  of  a  suitable  scale  for  any 
given  object,  it  has  been  thought  well  to  add  in  this  place 
the  following  Table  of  S<  ales,  adopted  for  the  guidance  of 


TOPOGRAPHICAL   DRAWING. 

TABLE  OF  SCALES. 


207 


PROPORTION  or  THE  SCALE. 


APPLICATION  or  THE  SCALE. 


1  inch  to  ^  an  incu, 

1  inch  to  1  inch, 
f 


1  inch  to  6  inches, 


1  inch  to  1  foot, 

A- 

1  inch  to  2  feet, 


1  inch  to  5  feet, 
irV 

1  inch  to  10  feet, 

rb- 

1  inch  to  50  feet, 
12  inches  to  200  yards, 

rb 

1  inch  to  220  feet, 
24  inches  to  1  mile, 


1  inch  to  440  feet, 
12  inches  to  1  mile. 


1  inch  to  880  feet, 
6  inches  to  1  mile. 

TOiTTT' 

1  inch  to  1320  feet, 
4  inches  to  1  mile, 


1  inch  to  2640  feet, 

2  inches  to  1  mile, 


1  inch  to  5280  feet, 
1  inch  to  1  mile, 


I  inch  to  10560  feet. 
£  an  inch  to  1  mile, 


Details  of  surveying  instruments,  <fcc., 
when  great  accuracy  is  required. 

All  models  for  masons,  carpenters, 
<kc. ;  and  for  the  drawings  of  small 
objects  requiring  the  details  accu- 
rately. 

Machines  and  tools  of  small  dimen- 
sions, as  the  jack,  axes,  saws,  <fec.; 
hangings  of  gates,  <fec.,  &c. 

Machines  of  mean  size,  as  capstans, 
windlasses,  vehicles  of  transporta- 
tion, &c. 

Large  machines,  as  pile  engines, 
pumps,  &c.;  details  of  arrange- 
ment of  stone  masonry,  of  car- 
pentry, <fec. 

Canal  lock,  scaffoldings,  separate 
drawings  of  light-houses,  buildings 
of  various  kinds. 

Vreiieral  plans  of  buildings  where 
minute  details  are  not  put  down. 

Sections  and  profiles  of  roads,  canals, 
<fec.  Maps  of  ground  with  hori- 
zontal curves  one  foot  apart. 

Topographical  maps  comprising  one 
mile  and  a  half  square,  as  important 
parts  of  anchorages,  harbors,  <fec. 

Topographical  maps  embracing  three 
miles  square. 


Topographical   maps   exceeding 
-and  within  eight  miles  square. 


four 


Topographical   maps  embracing  nine 
miles  square. 

Maps  not  exceeding  24  miles  square. 
Maps  comprising  50  miles  square. 
Maps  comprising  100  miles  square. 


208  INDUSTRIAL   DRAWING. 

the  Corps  of  Engineers.  The  first  column  of  this  table  giveg 
the  unit  of  the  drawing  which  corresponds  to  the  number 
of  units  of  the  object  itself;  and  under  this  the  vulgar 
fraction  which  shows  the  ratio  of  the  linear  dimensions  of  the 
drawing  to  the  corresponding  linear  dimensions  of  the 
object.  For  example,  at  No.  8  of  the  Table  we  find  for  the 
designation  of  the  scale  "one  inch  to  50fiet,  or  12  inches  to 
200  yards,"  and  below  this  the  fraction  yiff ;  this  then  is 
understood  to  express  that  the  distance  between  any  two 
points  on  the  drawing  being  one  inch  the  actual  horizontal 
distance  between  the  same  points  on  the  object  is  50  feet ;  or, 
if  the  distance  between  two  points  on  the  drawing  is  12 
inches  then  the  corresponding  horizontal  distance  on  the 
object  is  200  yards.  In  like  manner  that  every  linear  inch 
or  foot  on  the  drawing  is  the  ^1^  part  of  the  corresponding 
horizontal  distance  on  the  object. 

Copying  maps,  &c.  As  the  topographical  sketches  taken 
on  the  ground  are  to  serve  as  models  from  which  the 
finished  drawings  are  to  be  made,  either  on  the  same  scale  as 
the  sketch,  or  on  one  greater  or  smaller  than  it,  the  drafts- 
man must  have  some  accurate  and  at  the  same  time  speedy 
method  of  copying  from  an  original,  either  on  the  same,  or  a 
different  scale.  The  most  simple,  and  at  the  same  time  the 
moft  accurate  and  speedy  in  skilful  hands,  is  to  divide  the 
original  into  squares,  by  pencil  lines  drawn  on  it  lengthwise 
and  crosswise ;  the  side  of  the  square  being  of  any  convenient 
length  that  will  subserve  the  purposes  of  accuracy.  Having 
prepared  the  original  in  this  manner,  the  sheet  on  which  the 
copy  is  to  be  made  is  divided  into  squares  in  like  manner ; 
the  side  of  each  being  the  same  as  that  of  the  original  when 
the  copy  is  to  be  of  the  same  size;  or  in  any  proportion 
shorter  or  longer  than  those  of  the  original  when  the  linear 
dimensions  of  the  copy  are  to  be  in  any  given  proportion 
shorter  or  longer  than  those  of  the  original.  Having  pre 
pared  the  blank  sheet  in  this  way,  we  have  only  to  judge  by 
the  eye,  when  it  is  well  trained,  how  the  different  objects  on 
the  original  lie  with  respect  to  the  sides  of  the  squares  on  it ; 
to  be  enabled,  by  the  hand,  to  put  them  in  pencil  on  the 
blank  sheet.  After  roughly  putting  in  the  outline  work  ID 


TOPOGRAPHICAL   DRAWING.  209 

this  way,  the  corrections  of  inaccuracies  can  be  afterwards 
readily  effected.  All  the  outline  having  been  completed  in 
pencil,  the  labor  of  the  pen  is  commenced,  and  the  details, 
as  lines  of  declivity,  conventional  signs,  &c.,  are  put  in  bv  it 
alone. 


THE   END 


• 


University  of  California 

SOUTHERN  REGIONAL  LIBRARY  FACILITY 

Return  this  material  to  the  library 

from  which  it  was  borrowed. 


2  4 1990 
OCT191990 


Library 


:    310N 
1V1S 


u 


